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Assign value to an individual cell in a two dimensional python array [duplicate]

Let's say I have the following empty two dimensional array in Python:

I want to assign a value of 5 to the first row in the first column of q . Instinctively, I do the following:

However, this produces:

The first element of every array is being initialized to 5 , where I thought only the first element of the first array would get the update. I have two questions:

Why is Python initializing the first value of every array and not just the first one?

Is there a better way to accomplish what I'm trying to do?

• 1 same problem here, totally forgot about it use the reference of the object –  plasma Commented Jul 30, 2020 at 13:48

7 Answers 7

This doesn't do what you hoped.

It reuses list objects multiple times. As you can see when you made a change to one cell, which was in a reused list object.

A single list with a value of [None] is used five times.

A single list with a value of [[None]*5] is used four times.

Might be more what you're looking for.

This explicitly avoids reusing a list object.

80% of the time, a dictionary is what you really wanted.

Will also work. You don't start with a pre-defined grid of None values. But it's rare to need them in the first place.

In Python 2.7 and higher, you can do this.

That will build a grid indexed by 2-tuples.

• 7 I'll give you +1 if you share the research that led to that value of 80%. –  Niklas B. Commented Feb 27, 2012 at 2:38
• 9 It's the Pareto Principle. en.wikipedia.org/wiki/Pareto_principle . 80% of the time, you only need 20% of the data structures. –  S.Lott Commented Feb 27, 2012 at 2:49
• The same isn't exactly true for primitive type. The outer "4" is creating 4 references that are pointing to the same list, but the inner "5" isn't creating 5 references that are pointing to the same "None". So you can still do [[None] * 5 for j in range(4)] –  qkhhly Commented Sep 26, 2014 at 15:28
• 2 @qkhhly: No, it behaves exactly the same for all types. You can confirm that all references point to the same None by looking at [[id(x) for x in sublist] for sublist in q] . –  DSM Commented Nov 11, 2015 at 5:36
• Using list comprehension [[x for x in mylist] for x in range(9)] didn't help me in case of list of lists. Solved the problem using [[x for x in copy.deepcopy(mylist)] for x in range(9)] instead. –  Semen Commented Feb 24, 2021 at 10:07

The reason why is you have the list, just duplicated four times! Python isn't regenerating that list every time when you do *4 . It's using the same list object.

To get around this, you need for force python to regenrate that list for you every time:

In this case, I'm using a list comprehension.

The answer is simple Never use

as when you do assignment

q[0][1]=5 it assigns value multiple time to multiple rows at 1 column try print(q)

then q[0][1]=5 will assign one time only try

Because they are the same array, referred to multiple times.

Create the structure such that the outer array refers to separate inner arrays instead of reusing one. The other answers provide ways to do so.

The answer to question number 2: Using numpy is an option. See following code.

In case you want to use a list and not a dictionary as the others propose you can use this:

Not the answer you're looking for? Browse other questions tagged python arrays list or ask your own question .

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Nick McCullum

Software Developer & Professional Explainer

NumPy Indexing and Assignment

Hey - Nick here! This page is a free excerpt from my $199 course Python for Finance, which is 50% off for the next 50 students.

If you want the full course, click here to sign up.

In this lesson, we will explore indexing and assignment in NumPy arrays.

The Array I'll Be Using In This Lesson

As before, I will be using a specific array through this lesson. This time it will be generated using the np.random.rand method. Here's how I generated the array:

Here is the actual array:

To make this array easier to look at, I will round every element of the array to 2 decimal places using NumPy's round method:

Here's the new array:

How To Return A Specific Element From A NumPy Array

We can select (and return) a specific element from a NumPy array in the same way that we could using a normal Python list: using square brackets.

An example is below:

We can also reference multiple elements of a NumPy array using the colon operator. For example, the index [2:] selects every element from index 2 onwards. The index [:3] selects every element up to and excluding index 3. The index [2:4] returns every element from index 2 to index 4, excluding index 4. The higher endpoint is always excluded.

A few example of indexing using the colon operator are below.

Element Assignment in NumPy Arrays

We can assign new values to an element of a NumPy array using the = operator, just like regular python lists. A few examples are below (note that this is all one code block, which means that the element assignments are carried forward from step to step).

arr[2:5] = 0.5

Returns array([0. , 0. , 0.5, 0.5, 0.5])

As you can see, modifying second_new_array also changed the value of new_array .

Why is this?

By default, NumPy does not create a copy of an array when you reference the original array variable using the = assignment operator. Instead, it simply points the new variable to the old variable, which allows the second variable to make modification to the original variable - even if this is not your intention.

This may seem bizarre, but it does have a logical explanation. The purpose of array referencing is to conserve computing power. When working with large data sets, you would quickly run out of RAM if you created a new array every time you wanted to work with a slice of the array.

Fortunately, there is a workaround to array referencing. You can use the copy method to explicitly copy a NumPy array.

An example of this is below.

As you can see below, making modifications to the copied array does not alter the original.

So far in the lesson, we have only explored how to reference one-dimensional NumPy arrays. We will now explore the indexing of two-dimensional arrays.

Indexing Two-Dimensional NumPy Arrays

To start, let's create a two-dimensional NumPy array named mat :

There are two ways to index a two-dimensional NumPy array:

• mat[row, col]
• mat[row][col]

I personally prefer to index using the mat[row][col] nomenclature because it is easier to visualize in a step-by-step fashion. For example:

You can also generate sub-matrices from a two-dimensional NumPy array using this notation:

Array referencing also applies to two-dimensional arrays in NumPy, so be sure to use the copy method if you want to avoid inadvertently modifying an original array after saving a slice of it into a new variable name.

Conditional Selection Using NumPy Arrays

NumPy arrays support a feature called conditional selection , which allows you to generate a new array of boolean values that state whether each element within the array satisfies a particular if statement.

An example of this is below (I also re-created our original arr variable since its been awhile since we've seen it):

You can also generate a new array of values that satisfy this condition by passing the condition into the square brackets (just like we do for indexing).

An example of this is below:

Conditional selection can become significantly more complex than this. We will explore more examples in this section's associated practice problems.

In this lesson, we explored NumPy array indexing and assignment in thorough detail. We will solidify your knowledge of these concepts further by working through a batch of practice problems in the next section.

Structured arrays #

Introduction #.

Structured arrays are ndarrays whose datatype is a composition of simpler datatypes organized as a sequence of named fields . For example,

Here x is a one-dimensional array of length two whose datatype is a structure with three fields: 1. A string of length 10 or less named ‘name’, 2. a 32-bit integer named ‘age’, and 3. a 32-bit float named ‘weight’.

If you index x at position 1 you get a structure:

You can access and modify individual fields of a structured array by indexing with the field name:

Structured datatypes are designed to be able to mimic ‘structs’ in the C language, and share a similar memory layout. They are meant for interfacing with C code and for low-level manipulation of structured buffers, for example for interpreting binary blobs. For these purposes they support specialized features such as subarrays, nested datatypes, and unions, and allow control over the memory layout of the structure.

Users looking to manipulate tabular data, such as stored in csv files, may find other pydata projects more suitable, such as xarray, pandas, or DataArray. These provide a high-level interface for tabular data analysis and are better optimized for that use. For instance, the C-struct-like memory layout of structured arrays in numpy can lead to poor cache behavior in comparison.

Structured datatypes #

A structured datatype can be thought of as a sequence of bytes of a certain length (the structure’s itemsize ) which is interpreted as a collection of fields. Each field has a name, a datatype, and a byte offset within the structure. The datatype of a field may be any numpy datatype including other structured datatypes, and it may also be a subarray data type which behaves like an ndarray of a specified shape. The offsets of the fields are arbitrary, and fields may even overlap. These offsets are usually determined automatically by numpy, but can also be specified.

Structured datatype creation #

Structured datatypes may be created using the function numpy.dtype . There are 4 alternative forms of specification which vary in flexibility and conciseness. These are further documented in the Data Type Objects reference page, and in summary they are:

A list of tuples, one tuple per field

Each tuple has the form (fieldname, datatype, shape) where shape is optional. fieldname is a string (or tuple if titles are used, see Field Titles below), datatype may be any object convertible to a datatype, and shape is a tuple of integers specifying subarray shape.

If fieldname is the empty string '' , the field will be given a default name of the form f# , where # is the integer index of the field, counting from 0 from the left:

The byte offsets of the fields within the structure and the total structure itemsize are determined automatically.

A string of comma-separated dtype specifications

In this shorthand notation any of the string dtype specifications may be used in a string and separated by commas. The itemsize and byte offsets of the fields are determined automatically, and the field names are given the default names f0 , f1 , etc.

A dictionary of field parameter arrays

This is the most flexible form of specification since it allows control over the byte-offsets of the fields and the itemsize of the structure.

The dictionary has two required keys, ‘names’ and ‘formats’, and four optional keys, ‘offsets’, ‘itemsize’, ‘aligned’ and ‘titles’. The values for ‘names’ and ‘formats’ should respectively be a list of field names and a list of dtype specifications, of the same length. The optional ‘offsets’ value should be a list of integer byte-offsets, one for each field within the structure. If ‘offsets’ is not given the offsets are determined automatically. The optional ‘itemsize’ value should be an integer describing the total size in bytes of the dtype, which must be large enough to contain all the fields.

Offsets may be chosen such that the fields overlap, though this will mean that assigning to one field may clobber any overlapping field’s data. As an exception, fields of numpy.object_ type cannot overlap with other fields, because of the risk of clobbering the internal object pointer and then dereferencing it.

The optional ‘aligned’ value can be set to True to make the automatic offset computation use aligned offsets (see Automatic byte offsets and alignment ), as if the ‘align’ keyword argument of numpy.dtype had been set to True.

The optional ‘titles’ value should be a list of titles of the same length as ‘names’, see Field Titles below.

A dictionary of field names

The keys of the dictionary are the field names and the values are tuples specifying type and offset:

This form was discouraged because Python dictionaries did not preserve order in Python versions before Python 3.6. Field Titles may be specified by using a 3-tuple, see below.

Manipulating and displaying structured datatypes #

The list of field names of a structured datatype can be found in the names attribute of the dtype object:

The dtype of each individual field can be looked up by name:

The field names may be modified by assigning to the names attribute using a sequence of strings of the same length.

The dtype object also has a dictionary-like attribute, fields , whose keys are the field names (and Field Titles , see below) and whose values are tuples containing the dtype and byte offset of each field.

Both the names and fields attributes will equal None for unstructured arrays. The recommended way to test if a dtype is structured is with if dt.names is not None rather than if dt.names , to account for dtypes with 0 fields.

The string representation of a structured datatype is shown in the “list of tuples” form if possible, otherwise numpy falls back to using the more general dictionary form.

Automatic byte offsets and alignment #

Numpy uses one of two methods to automatically determine the field byte offsets and the overall itemsize of a structured datatype, depending on whether align=True was specified as a keyword argument to numpy.dtype .

By default ( align=False ), numpy will pack the fields together such that each field starts at the byte offset the previous field ended, and the fields are contiguous in memory.

If align=True is set, numpy will pad the structure in the same way many C compilers would pad a C-struct. Aligned structures can give a performance improvement in some cases, at the cost of increased datatype size. Padding bytes are inserted between fields such that each field’s byte offset will be a multiple of that field’s alignment, which is usually equal to the field’s size in bytes for simple datatypes, see PyArray_Descr.alignment . The structure will also have trailing padding added so that its itemsize is a multiple of the largest field’s alignment.

Note that although almost all modern C compilers pad in this way by default, padding in C structs is C-implementation-dependent so this memory layout is not guaranteed to exactly match that of a corresponding struct in a C program. Some work may be needed, either on the numpy side or the C side, to obtain exact correspondence.

If offsets were specified using the optional offsets key in the dictionary-based dtype specification, setting align=True will check that each field’s offset is a multiple of its size and that the itemsize is a multiple of the largest field size, and raise an exception if not.

If the offsets of the fields and itemsize of a structured array satisfy the alignment conditions, the array will have the ALIGNED flag set.

A convenience function numpy.lib.recfunctions.repack_fields converts an aligned dtype or array to a packed one and vice versa. It takes either a dtype or structured ndarray as an argument, and returns a copy with fields re-packed, with or without padding bytes.

Field titles #

In addition to field names, fields may also have an associated title , an alternate name, which is sometimes used as an additional description or alias for the field. The title may be used to index an array, just like a field name.

To add titles when using the list-of-tuples form of dtype specification, the field name may be specified as a tuple of two strings instead of a single string, which will be the field’s title and field name respectively. For example:

When using the first form of dictionary-based specification, the titles may be supplied as an extra 'titles' key as described above. When using the second (discouraged) dictionary-based specification, the title can be supplied by providing a 3-element tuple (datatype, offset, title) instead of the usual 2-element tuple:

The dtype.fields dictionary will contain titles as keys, if any titles are used. This means effectively that a field with a title will be represented twice in the fields dictionary. The tuple values for these fields will also have a third element, the field title. Because of this, and because the names attribute preserves the field order while the fields attribute may not, it is recommended to iterate through the fields of a dtype using the names attribute of the dtype, which will not list titles, as in:

Union types #

Structured datatypes are implemented in numpy to have base type numpy.void by default, but it is possible to interpret other numpy types as structured types using the (base_dtype, dtype) form of dtype specification described in Data Type Objects . Here, base_dtype is the desired underlying dtype, and fields and flags will be copied from dtype . This dtype is similar to a ‘union’ in C.

Indexing and assignment to structured arrays #

Assigning data to a structured array #.

There are a number of ways to assign values to a structured array: Using python tuples, using scalar values, or using other structured arrays.

Assignment from Python Native Types (Tuples) #

The simplest way to assign values to a structured array is using python tuples. Each assigned value should be a tuple of length equal to the number of fields in the array, and not a list or array as these will trigger numpy’s broadcasting rules. The tuple’s elements are assigned to the successive fields of the array, from left to right:

Assignment from Scalars #

A scalar assigned to a structured element will be assigned to all fields. This happens when a scalar is assigned to a structured array, or when an unstructured array is assigned to a structured array:

Structured arrays can also be assigned to unstructured arrays, but only if the structured datatype has just a single field:

Assignment from other Structured Arrays #

Assignment between two structured arrays occurs as if the source elements had been converted to tuples and then assigned to the destination elements. That is, the first field of the source array is assigned to the first field of the destination array, and the second field likewise, and so on, regardless of field names. Structured arrays with a different number of fields cannot be assigned to each other. Bytes of the destination structure which are not included in any of the fields are unaffected.

Assignment involving subarrays #

When assigning to fields which are subarrays, the assigned value will first be broadcast to the shape of the subarray.

Indexing structured arrays #

Accessing individual fields #.

Individual fields of a structured array may be accessed and modified by indexing the array with the field name.

The resulting array is a view into the original array. It shares the same memory locations and writing to the view will modify the original array.

This view has the same dtype and itemsize as the indexed field, so it is typically a non-structured array, except in the case of nested structures.

If the accessed field is a subarray, the dimensions of the subarray are appended to the shape of the result:

Accessing Multiple Fields #

One can index and assign to a structured array with a multi-field index, where the index is a list of field names.

The behavior of multi-field indexes changed from Numpy 1.15 to Numpy 1.16.

The result of indexing with a multi-field index is a view into the original array, as follows:

Assignment to the view modifies the original array. The view’s fields will be in the order they were indexed. Note that unlike for single-field indexing, the dtype of the view has the same itemsize as the original array, and has fields at the same offsets as in the original array, and unindexed fields are merely missing.

In Numpy 1.15, indexing an array with a multi-field index returned a copy of the result above, but with fields packed together in memory as if passed through numpy.lib.recfunctions.repack_fields .

The new behavior as of Numpy 1.16 leads to extra “padding” bytes at the location of unindexed fields compared to 1.15. You will need to update any code which depends on the data having a “packed” layout. For instance code such as:

will need to be changed. This code has raised a FutureWarning since Numpy 1.12, and similar code has raised FutureWarning since 1.7.

In 1.16 a number of functions have been introduced in the numpy.lib.recfunctions module to help users account for this change. These are numpy.lib.recfunctions.repack_fields . numpy.lib.recfunctions.structured_to_unstructured , numpy.lib.recfunctions.unstructured_to_structured , numpy.lib.recfunctions.apply_along_fields , numpy.lib.recfunctions.assign_fields_by_name , and numpy.lib.recfunctions.require_fields .

The function numpy.lib.recfunctions.repack_fields can always be used to reproduce the old behavior, as it will return a packed copy of the structured array. The code above, for example, can be replaced with:

Furthermore, numpy now provides a new function numpy.lib.recfunctions.structured_to_unstructured which is a safer and more efficient alternative for users who wish to convert structured arrays to unstructured arrays, as the view above is often intended to do. This function allows safe conversion to an unstructured type taking into account padding, often avoids a copy, and also casts the datatypes as needed, unlike the view. Code such as:

can be made safer by replacing with:

Assignment to an array with a multi-field index modifies the original array:

This obeys the structured array assignment rules described above. For example, this means that one can swap the values of two fields using appropriate multi-field indexes:

Indexing with an Integer to get a Structured Scalar #

Indexing a single element of a structured array (with an integer index) returns a structured scalar:

Unlike other numpy scalars, structured scalars are mutable and act like views into the original array, such that modifying the scalar will modify the original array. Structured scalars also support access and assignment by field name:

Similarly to tuples, structured scalars can also be indexed with an integer:

Thus, tuples might be thought of as the native Python equivalent to numpy’s structured types, much like native python integers are the equivalent to numpy’s integer types. Structured scalars may be converted to a tuple by calling numpy.ndarray.item :

Viewing structured arrays containing objects #

In order to prevent clobbering object pointers in fields of object type, numpy currently does not allow views of structured arrays containing objects.

Structure comparison and promotion #

If the dtypes of two void structured arrays are equal, testing the equality of the arrays will result in a boolean array with the dimensions of the original arrays, with elements set to True where all fields of the corresponding structures are equal:

NumPy will promote individual field datatypes to perform the comparison. So the following is also valid (note the 'f4' dtype for the 'a' field):

To compare two structured arrays, it must be possible to promote them to a common dtype as returned by numpy.result_type and numpy.promote_types . This enforces that the number of fields, the field names, and the field titles must match precisely. When promotion is not possible, for example due to mismatching field names, NumPy will raise an error. Promotion between two structured dtypes results in a canonical dtype that ensures native byte-order for all fields:

The resulting dtype from promotion is also guaranteed to be packed, meaning that all fields are ordered contiguously and any unnecessary padding is removed:

Note that the result prints without offsets or itemsize indicating no additional padding. If a structured dtype is created with align=True ensuring that dtype.isalignedstruct is true, this property is preserved:

When promoting multiple dtypes, the result is aligned if any of the inputs is:

The < and > operators always return False when comparing void structured arrays, and arithmetic and bitwise operations are not supported.

Changed in version 1.23: Before NumPy 1.23, a warning was given and False returned when promotion to a common dtype failed. Further, promotion was much more restrictive: It would reject the mixed float/integer comparison example above.

Record arrays #

As an optional convenience numpy provides an ndarray subclass, numpy.recarray that allows access to fields of structured arrays by attribute instead of only by index. Record arrays use a special datatype, numpy.record , that allows field access by attribute on the structured scalars obtained from the array. The numpy.rec module provides functions for creating recarrays from various objects. Additional helper functions for creating and manipulating structured arrays can be found in numpy.lib.recfunctions .

The simplest way to create a record array is with numpy.rec.array :

numpy.rec.array can convert a wide variety of arguments into record arrays, including structured arrays:

The numpy.rec module provides a number of other convenience functions for creating record arrays, see record array creation routines .

A record array representation of a structured array can be obtained using the appropriate view :

For convenience, viewing an ndarray as type numpy.recarray will automatically convert to numpy.record datatype, so the dtype can be left out of the view:

To get back to a plain ndarray both the dtype and type must be reset. The following view does so, taking into account the unusual case that the recordarr was not a structured type:

Record array fields accessed by index or by attribute are returned as a record array if the field has a structured type but as a plain ndarray otherwise.

Note that if a field has the same name as an ndarray attribute, the ndarray attribute takes precedence. Such fields will be inaccessible by attribute but will still be accessible by index.

Recarray helper functions #

Collection of utilities to manipulate structured arrays.

Most of these functions were initially implemented by John Hunter for matplotlib. They have been rewritten and extended for convenience.

Add new fields to an existing array.

The names of the fields are given with the names arguments, the corresponding values with the data arguments. If a single field is appended, names , data and dtypes do not have to be lists but just values.

Input array to extend.

String or sequence of strings corresponding to the names of the new fields.

Array or sequence of arrays storing the fields to add to the base.

Datatype or sequence of datatypes. If None, the datatypes are estimated from the data .

Filling value used to pad missing data on the shorter arrays.

Whether to return a masked array or not.

Whether to return a recarray (MaskedRecords) or not.

Apply function ‘func’ as a reduction across fields of a structured array.

This is similar to numpy.apply_along_axis , but treats the fields of a structured array as an extra axis. The fields are all first cast to a common type following the type-promotion rules from numpy.result_type applied to the field’s dtypes.

Function to apply on the “field” dimension. This function must support an axis argument, like numpy.mean , numpy.sum , etc.

Structured array for which to apply func.

Result of the recution operation

Assigns values from one structured array to another by field name.

Normally in numpy >= 1.14, assignment of one structured array to another copies fields “by position”, meaning that the first field from the src is copied to the first field of the dst, and so on, regardless of field name.

This function instead copies “by field name”, such that fields in the dst are assigned from the identically named field in the src. This applies recursively for nested structures. This is how structure assignment worked in numpy >= 1.6 to <= 1.13.

The source and destination arrays during assignment.

If True, fields in the dst for which there was no matching field in the src are filled with the value 0 (zero). This was the behavior of numpy <= 1.13. If False, those fields are not modified.

Return a new array with fields in drop_names dropped.

Nested fields are supported.

Changed in version 1.18.0: drop_fields returns an array with 0 fields if all fields are dropped, rather than returning None as it did previously.

Input array

String or sequence of strings corresponding to the names of the fields to drop.

Whether to return a recarray or a mrecarray ( asrecarray=True ) or a plain ndarray or masked array with flexible dtype. The default is False.

Find the duplicates in a structured array along a given key

Name of the fields along which to check the duplicates. If None, the search is performed by records

Whether masked data should be discarded or considered as duplicates.

Whether to return the indices of the duplicated values.

Flatten a structured data-type description.

Returns a dictionary with fields indexing lists of their parent fields.

This function is used to simplify access to fields nested in other fields.

Input datatype

Last processed field name (used internally during recursion).

Dictionary of parent fields (used interbally during recursion).

Returns the field names of the input datatype as a tuple. Input datatype must have fields otherwise error is raised.

Returns the field names of the input datatype as a tuple. Input datatype must have fields otherwise error is raised. Nested structure are flattened beforehand.

Join arrays r1 and r2 on key key .

The key should be either a string or a sequence of string corresponding to the fields used to join the array. An exception is raised if the key field cannot be found in the two input arrays. Neither r1 nor r2 should have any duplicates along key : the presence of duplicates will make the output quite unreliable. Note that duplicates are not looked for by the algorithm.

A string or a sequence of strings corresponding to the fields used for comparison.

Structured arrays.

If ‘inner’, returns the elements common to both r1 and r2. If ‘outer’, returns the common elements as well as the elements of r1 not in r2 and the elements of not in r2. If ‘leftouter’, returns the common elements and the elements of r1 not in r2.

String appended to the names of the fields of r1 that are present in r2 but absent of the key.

String appended to the names of the fields of r2 that are present in r1 but absent of the key.

Dictionary mapping field names to the corresponding default values.

Whether to return a MaskedArray (or MaskedRecords is asrecarray==True ) or a ndarray.

Whether to return a recarray (or MaskedRecords if usemask==True ) or just a flexible-type ndarray.

The output is sorted along the key.

A temporary array is formed by dropping the fields not in the key for the two arrays and concatenating the result. This array is then sorted, and the common entries selected. The output is constructed by filling the fields with the selected entries. Matching is not preserved if there are some duplicates…

Merge arrays field by field.

Sequence of arrays

Whether to collapse nested fields.

Without a mask, the missing value will be filled with something, depending on what its corresponding type:

-1 for integers

-1.0 for floating point numbers

'-' for characters

'-1' for strings

True for boolean values

XXX: I just obtained these values empirically

Returns a new numpy.recarray with fields in drop_names dropped.

Join arrays r1 and r2 on keys. Alternative to join_by, that always returns a np.recarray.

equivalent function

Fills fields from output with fields from input, with support for nested structures.

Input array.

Output array.

output should be at least the same size as input

Rename the fields from a flexible-datatype ndarray or recarray.

Input array whose fields must be modified.

Dictionary mapping old field names to their new version.

Re-pack the fields of a structured array or dtype in memory.

The memory layout of structured datatypes allows fields at arbitrary byte offsets. This means the fields can be separated by padding bytes, their offsets can be non-monotonically increasing, and they can overlap.

This method removes any overlaps and reorders the fields in memory so they have increasing byte offsets, and adds or removes padding bytes depending on the align option, which behaves like the align option to numpy.dtype .

If align=False , this method produces a “packed” memory layout in which each field starts at the byte the previous field ended, and any padding bytes are removed.

If align=True , this methods produces an “aligned” memory layout in which each field’s offset is a multiple of its alignment, and the total itemsize is a multiple of the largest alignment, by adding padding bytes as needed.

array or dtype for which to repack the fields.

If true, use an “aligned” memory layout, otherwise use a “packed” layout.

If True, also repack nested structures.

Copy of a with fields repacked, or a itself if no repacking was needed.

Casts a structured array to a new dtype using assignment by field-name.

This function assigns from the old to the new array by name, so the value of a field in the output array is the value of the field with the same name in the source array. This has the effect of creating a new ndarray containing only the fields “required” by the required_dtype.

If a field name in the required_dtype does not exist in the input array, that field is created and set to 0 in the output array.

array to cast

datatype for output array

array with the new dtype, with field values copied from the fields in the input array with the same name

Superposes arrays fields by fields

Sequence of input arrays.

Whether automatically cast the type of the field to the maximum.

Converts an n-D structured array into an (n+1)-D unstructured array.

The new array will have a new last dimension equal in size to the number of field-elements of the input array. If not supplied, the output datatype is determined from the numpy type promotion rules applied to all the field datatypes.

Nested fields, as well as each element of any subarray fields, all count as a single field-elements.

Structured array or dtype to convert. Cannot contain object datatype.

The dtype of the output unstructured array.

If true, always return a copy. If false, a view is returned if possible, such as when the dtype and strides of the fields are suitable and the array subtype is one of numpy.ndarray , numpy.recarray or numpy.memmap .

Changed in version 1.25.0: A view can now be returned if the fields are separated by a uniform stride.

See casting argument of numpy.ndarray.astype . Controls what kind of data casting may occur.

Unstructured array with one more dimension.

Converts an n-D unstructured array into an (n-1)-D structured array.

The last dimension of the input array is converted into a structure, with number of field-elements equal to the size of the last dimension of the input array. By default all output fields have the input array’s dtype, but an output structured dtype with an equal number of fields-elements can be supplied instead.

Nested fields, as well as each element of any subarray fields, all count towards the number of field-elements.

Unstructured array or dtype to convert.

The structured dtype of the output array

If dtype is not supplied, this specifies the field names for the output dtype, in order. The field dtypes will be the same as the input array.

Whether to create an aligned memory layout.

See copy argument to numpy.ndarray.astype . If true, always return a copy. If false, and dtype requirements are satisfied, a view is returned.

Structured array with fewer dimensions.

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• Introduction
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Numpy Linear Algebra

A matrix is a two-dimensional data structure where numbers are arranged into rows and columns. For example,

The above matrix is a 3x3 (pronounced "three by three") matrix because it has 3 rows and 3 columns.

Here are some of the basic matrix operations provided by NumPy.

• Create Matrix in NumPy

In NumPy, we use the np.array() function to create a matrix. For example,

Here, we have created two matrices: 2x2 matrix and 3x3 matrix by passing a list of lists to the np.array() function respectively.

• Perform Matrix Multiplication in NumPy

We use the np.dot() function to perform multiplication between two matrices.

Let's see an example.

In this example, we have used the np.dot(matrix1, matrix2) function to perform matrix multiplication between two matrices: matrix1 and matrix2 .

To learn more about Matrix multiplication, please visit NumPy Matrix Multiplication .

Note : We can only take a dot product of matrices when they have a common dimension size. For example, For A = (M x N) and B = (N x K) when we take a dot product of C = A . B the resulting matrix is of size C = (M x K) .

• Transpose NumPy Matrix

The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. For 2x2 matrix,

In NumPy, we can obtain the transpose of a matrix using the np.transpose() function. For example,

Here, we have used the np.transpose(matrix1) function to obtain the transpose of matrix1 .

Note : Alternatively, we can use the .T attribute to get the transpose of a matrix. For example, if we used matrix1.T in our previous example, the result would be the same.

• Calculate Inverse of a Matrix in NumPy

In NumPy, we use the np.linalg.inv() function to calculate the inverse of the given matrix.

However, it is important to note that not all matrices have an inverse. Only square matrices that have a non-zero determinant have an inverse.

Now, let's use np.linalg.inv() to calculate the inverse of a square matrix.

Note : If we try to find the inverse of a non-square matrix, we will get an error message: numpy.linalg.linalgerror: Last 2 dimensions of the array must be square

Find Determinant of a Matrix in NumPy

We can find the determinant of a square matrix using the np.linalg.det() function to calculate the determinant of the given matrix.

Suppose we have a 2x2 matrix A :

So, the determinant of a 2x2 matrix will be:

where a, b, c, and d are the elements of the matrix.

Here, we have used the np.linalg.det(matrix1) function to find the determinant of the square matrix matrix1 .

• Flatten Matrix in NumPy

Flattening a matrix simply means converting a matrix into a 1D array.

To flatten a matrix into a 1-D array we use the array.flatten() function. Let's see an example.

Here, we have used the matrix1.flatten() function to flatten matrix1 into a 1D array, without compromising any of its elements

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Matrix manipulation in Python

In python matrix can be implemented as 2D list or 2D Array. Forming matrix from latter, gives the additional functionalities for performing various operations in matrix. These operations and array are defines in module “ numpy “. 

Operation on Matrix :  

• 1. add() :- This function is used to perform element wise matrix addition . 
• 2. subtract() :- This function is used to perform element wise matrix subtraction . 
• 3. divide() :- This function is used to perform element wise matrix division . 

Implementation:

• 4. multiply() :- This function is used to perform element wise matrix multiplication . 
• 5. dot() :- This function is used to compute the matrix multiplication, rather than element wise multiplication . 
• 6. sqrt() :- This function is used to compute the square root of each element of matrix. 
• 7. sum(x,axis) :- This function is used to add all the elements in matrix . Optional “axis” argument computes the column sum if axis is 0 and row sum if axis is 1 . 
• 8. “T” :- This argument is used to transpose the specified matrix. 

Using nested loops:

• Define matrices A and B.
• Get the number of rows and columns of the matrices using the len() function.
• Initialize matrices C, D, and E with zeros using nested loops or list comprehension.
• Use nested loops or list comprehension to perform the element-wise addition, subtraction, and division of matrices.
• Print the resulting matrices C, D, and E.

Time complexity: O(n^2) Space complexity: O(n^2)

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Linear Algebra in Python: Matrix Inverses and Least Squares

Table of Contents

Getting Started With Linear Algebra in Python

Understanding vectors, matrices, and the role of linear algebra, using determinants to study linear systems, using matrix inverses to solve linear systems, calculating inverses and determinants with scipy.linalg, interpolating polynomials with linear systems, building least squares models using scipy.linalg, obtaining least squares solutions using a pseudoinverse, preparing the data, building the model, predicting prices.

Linear algebra is an important topic across a variety of subjects. It allows you to solve problems related to vectors , matrices , and linear equations . In Python, most of the routines related to this subject are implemented in scipy.linalg , which offers very fast linear algebra capabilities.

In particular, linear models play an important role in a variety of real-world problems, and scipy.linalg provides tools to compute them in an efficient way.

In this tutorial, you’ll learn how to:

• Study linear systems using determinants and solve problems using matrix inverses
• Interpolate polynomials to fit a set of points using linear systems
• Use Python to solve linear regression problems
• Use linear regression to predict prices based on historical data

This is the second part of a series of tutorials on linear algebra using scipy.linalg . So, before continuing, make sure to take a look at the first tutorial of the series before reading this one.

Free Source Code: Click here to download the free code and dataset that you’ll use to work with linear systems and algebra in Python with scipy.linalg .

Now you’re ready to get started!

Linear algebra is a branch of mathematics that deals with linear equations and their representations using vectors and matrices . It’s a fundamental subject in several areas of engineering, and it’s a prerequisite to a deeper understanding of machine learning .

To work with linear algebra in Python, you can count on SciPy , which is an open-source Python library used for scientific computing, including several modules for common tasks in science and engineering.

Of course, SciPy includes modules for linear algebra , but that’s not all. It also offers optimization , integration , interpolation , and signal processing capabilities. It’s part of the SciPy stack , which includes several other packages for scientific computing, such as NumPy , Matplotlib , SymPy , IPython , and pandas .

scipy.linalg includes several tools for working with linear algebra problems, including functions for performing matrix calculations, such as determinants , inverses , eigenvalues, eigenvectors , and the singular value decomposition .

In the previous tutorial of this series , you learned how to work with matrices and vectors in Python to model practical problems using linear systems. You solved these problems using scipy.linalg .

In this tutorial, you’re going a step further, using scipy.linalg to study linear systems and build linear models for real-world problems.

In order to use scipy.linalg , you have to install and set up the SciPy library. Besides that, you’re going to use Jupyter Notebook to run the code in an interactive environment. SciPy and Jupyter Notebook are third-party packages that you need to install. For installation, you can use the conda or pip package manager. Revisit Working With Linear Systems in Python With scipy.linalg for installation details.

Note: Using Jupyter Notebook to run the code isn’t mandatory, but it facilitates working with numerical and scientific applications.

For a refresher on working with Jupyter Notebooks, take a look at Jupyter Notebook: An Introduction .

Next, you’ll go through some fundamental concepts of linear algebra and explore how to use Python to work with these concepts.

A vector is a mathematical entity used to represent physical quantities that have both magnitude and direction. It’s a fundamental tool for solving engineering and machine learning problems. So are matrices , which are used to represent vector transformations, among other applications.

Note: In Python, NumPy is the most used library for working with matrices and vectors. It uses a special type called ndarray to represent them. As an example, imagine that you need to create the following matrix:

With NumPy, you can use np.array() to create it, providing a nested list containing the elements of each row of the matrix:

NumPy provides several functions to facilitate working with vector and matrix computations. You can find more information on how to use NumPy to represent vectors and matrices and perform operations with them in the previous tutorial in this series .

A linear system or, more precisely, a system of linear equations, is a set of equations linearly relating to a set of variables. Here’s an example of a linear system relating to the variables x ₁ and x ₂:

Here you have two equations involving two variables. In order to have a linear system, the values that multiply the variables x ₁ and x ₂ must be constants, like the ones in this example. It’s common to write linear systems using matrices and vectors. For example, you can write the previous system as the following matrix product :

Comparing the matrix product form with the original system, you can notice the elements of matrix A correspond to the coefficients that multiply x ₁ and x ₂. Besides that, the values in the right-hand side of the original equations now make up vector b .

Linear algebra is a mathematical discipline that deals with vectors, matrices, and vector spaces and linear transformations more generally. By using linear algebra concepts, it’s possible to build algorithms to perform computations for several applications, including solving linear systems.

When there are just two or three equations and variables, it’s feasible to perform the calculations manually , combine the equations, and find the values for the variables.

However, in real-world applications, the number of equations can be very large, making it infeasible to do calculations manually. That’s precisely when linear algebra concepts and algorithms come handy, allowing you to develop usable applications for engineering and machine learning , for example.

In Working With Linear Systems in Python With scipy.linalg , you’ve seen how to solve linear systems using scipy.linalg.solve() . Now you’re going to learn how to use determinants to study the possible solutions and how to solve problems using the concept of matrix inverses.

Solving Problems Using Matrix Inverses and Determinants

Matrix inverses and determinants are tools that allow you to get some information about the linear system and also to solve it. Before going through the details on how to calculate matrix inverses and determinants using scipy.linalg , take some time to remember how to use these structures.

As you may recall from your math classes, not every linear system can be solved. You may have a combination of equations that’s inconsistent and has no solution. For example, a system with two equations given by x ₁ + x ₂ = 2 and x ₁ + x ₂ = 3 is inconsistent and has no solution. This happens because no two numbers x ₁ and x ₂ can add up to both 2 and 3 at the same time.

Besides that, some systems can be solved but have more than one solution. For example, if you have a system with two equivalent equations, such as x ₁ + x ₂ = 2 and 2 x ₁ + 2 x ₂ = 4, then you can find an infinite number of solutions, such as ( x ₁=1, x ₂=1), ( x ₁=0, x ₂=2), ( x ₁=2, x ₂=0), and so on.

A determinant is a number, calculated using the matrix of coefficients , that tells you if there’s a solution for the system. Because you’ll be using scipy.linalg to calculate it, you don’t need to care much about the details on how to make the calculation. However, keep the following in mind:

• If the determinant of a coefficients matrix of a linear system is different from zero , then you can say the system has a unique solution .
• If the determinant of a coefficients matrix of a linear system is equal to zero , then the system may have either zero solutions or an infinite number of solutions .

Now that you have this in mind, you’ll learn how to solve linear systems using matrices.

To understand the idea behind the inverse of a matrix, start by recalling the concept of the multiplicative inverse of a number. When you multiply a number by its inverse, you get 1 as the result. Take 3 as an example. The inverse of 3 is 1/3, and when you multiply these numbers, you get 3 × 1/3 = 1.

With square matrices, you can think of a similar idea. However, instead of 1, you’ll get an identity matrix as the result. An identity matrix has ones in its diagonal and zeros in the elements outside of the diagonal, like the following examples:

The identity matrix has an interesting property: when multiplied by another matrix A of the same dimensions, the obtained result is A . Recall that this is also true for the number 1, when you consider the multiplication of numbers.

This allows you to solve a linear system by following the same steps used to solve an equation. As an example, consider the following linear system, written as a matrix product:

By calling A ⁻¹ the inverse of matrix A , you could multiply both sides of the equation by A ⁻¹, which would give you the following result:

This way, by using the inverse, A ⁻¹, you can obtain the solution x for the system by calculating A ⁻¹ b .

It’s worth noting that while non-zero numbers always have an inverse, not all matrices have an inverse. When the system has no solution or when it has multiple solutions, the determinant of A will be zero, and the inverse, A ⁻¹, won’t exist.

Now you’ll see how to use Python with scipy.linalg to make these calculations.

You can calculate matrix inverses and determinants using scipy.linalg.inv() and scipy.linalg.det() .

For example, consider the meal plan problem that you worked on in the previous tutorial of this series . Recall that the linear system for this problem could be written as a matrix product:

Previously, you used scipy.linalg.solve() to obtain the solution 10 , 10 , 20 , 20 , 10 for the variables x ₁ to x ₅, respectively. But as you’ve just learned, it’s also possible to use the inverse of the coefficients matrix to obtain vector x , which contains the solutions for the problem. You have to calculate x = A ⁻¹ b , which you can do with the following program:

Here’s a breakdown of what’s happening:

Lines 1 and 2 import NumPy as np , along with linalg from scipy . These imports allow you to use linalg.inv() .

Lines 4 to 12 create the coefficients matrix as a NumPy array called A .

Line 14 creates the independent terms vector as a NumPy array called b . To make it a column vector with five elements, you use .reshape((5, 1)) .

Line 16 uses linalg.inv() to obtain the inverse of matrix A .

Lines 18 and 19 use the @ operator to perform the matrix product in order to solve the linear system characterized by A and b . You store the result in x , which is printed.

You get exactly the same solution as the one provided by scipy.linalg.solve() . Because this system has a unique solution, the determinant of matrix A must be different from zero. You can confirm that it is by calculating it using det() from scipy.linalg :

As expected, the determinant isn’t zero. This indicates that the inverse of A , denoted as A ⁻¹ and calculated with inv(A) , exists, so the system has a unique solution. A ⁻¹ is a square matrix with the same dimensions as A , so the product of A ⁻¹ and A results in an identity matrix. In this example, it’s given by the following:

Now that you know the basics of using matrix inverses and determinants, you’ll see how to use these tools to find the coefficients of polynomials.

You can use linear systems to calculate polynomial coefficients so that these polynomials include some specific points.

For example, consider the second-degree polynomial y = P ( x ) = a ₀ + a ₁ x + a ₂ x ². Recall that when you plot a second-degree polynomial, you get a parabola, which will be different depending on the coefficients a ₀, a ₁, and a ₂.

Now, suppose that you’d like to find a specific second-degree polynomial that includes the ( x , y ) points (1, 5), (2, 13), and (3, 25). How could you calculate a ₀, a ₁, and a ₂, such that P ( x ) includes these points in its parabola? In other words, you want to find the coefficients of the polynomial in this figure:

For each point that you’d like to include in the parabola, you can use the general expression of the polynomial in order to get a linear equation. For example, taking the second point, ( x =2, y =13), and considering that y = a ₀ + a ₁ x + a ₂ x ², you could write the following equation:

This way, for each point ( x , y ), you’ll get an equation involving a ₀, a ₁, and a ₂. Because you’re considering three different points, you’ll end up with a system of three equations:

To check if this system has a unique solution, you can calculate the determinant of the coefficients matrix and check if it’s not zero. You can do that with the following code:

It’s worth noting that the existence of the solution only depends on A . Because the value of the determinant isn’t zero, you can be sure that there’s a unique solution for the system. You can solve it using the matrix inverse method with the following code:

This result tells you that a ₀ = 1, a ₁ = 2, and a ₂ = 2 is a solution for the system. In other words, the polynomial that includes the points (1, 5), (2, 13), and (3, 25) is given by y = P ( x ) = 1 + 2 x + 2 x ². You can test the solution for each point by inputting x and verifying that P ( x ) is equal to y .

As an example of a system without any solution, say that you’re trying to interpolate a parabola with the ( x , y ) points given by (1, 5), (2, 13), and (2, 25). If you look carefully at these numbers, you’ll notice that the second and third points consider x = 2 and different values for y , which makes it impossible to find a function that includes both points.

Following the same steps as before, you’ll arrive at the equations for this system, which are the following:

To confirm that this system doesn’t present a unique solution, you can calculate the determinant of the coefficients matrix with the following code:

You may notice that the value of the determinant is zero, which means that the system doesn’t have a unique solution. This also means that the inverse of the coefficients matrix doesn’t exist. In other words, the coefficients matrix is singular .

Depending on your computer architecture, you may get a very small number instead of zero. This happens due to the numerical algorithms that det() uses to calculate the determinant. In these algorithms, numeric precision errors make this result not exactly equal to zero.

In general, whenever you come across a tiny number, you can conclude that the system doesn’t have a unique solution.

You can try to solve the linear system using the matrix inverse method with the following code:

Because the system has no solution, you get an exception telling you that the coefficients matrix is singular.

When the system has more than one solution, you’ll come across a similar result. The value of the determinant of the coefficients matrix will be zero or very small, indicating that the coefficients matrix again is singular.

As an example of a system with more than one solution, you can try to interpolate a parabola considering the points ( x , y ) given by (1, 5), (2, 13), and (2, 13). As you may notice, here you’re considering two points at the same position, which allows an infinite number of solutions for a ₀, a ₁, and a ₂.

Now that you’ve gone through how to work with polynomial interpolation using linear systems, you’ll see another technique that makes an effort to find the coefficients for any set of points.

Minimizing Error With Least Squares

You’ve seen that sometimes you can’t find a polynomial that fits precisely to a set of points. However, usually when you’re trying to interpolate a polynomial, you’re not interested in a precise fit. You’re just looking for a solution that approximates the points, providing the minimum error possible.

This is generally the case when you’re working with real-world data. Usually, it includes some noise caused by errors that occur in the collecting process, like imprecision or malfunction in sensors, and typos when users are inputting data manually.

Using the least squares method, you can find a solution for the interpolation of a polynomial, even when the coefficients matrix is singular. By using this method, you’ll be looking for the coefficients of the polynomial that provides the minimum squared error when comparing the polynomial curve to your data points.

Actually, the least squares method is generally used to fit polynomials to large sets of data points. The idea is to try to design a model that represents some observed behavior.

Note: If a linear system has a unique solution, then the least squares solution will be equal to that unique solution.

For example, you could design a model to try to predict car prices. For that, you could collect some real-world data, including the car price and some other features like the mileage, the year, and the type of car. With this data, you can design a polynomial that models the price as a function of the other features and use least squares to find the optimal coefficients of this model.

Soon, you’re going to work on a model to address this problem. But first, you’re going to see how to use scipy.linalg to build models using least squares.

To solve least squares problems, scipy.linalg provides a function called lstsq() . To see how it works, consider the previous example, in which you tried to fit a parabola to the points ( x , y ) given by (1, 5), (2, 13), and (2, 25). Remember that this system has no solution, since there are two points with the same value for x .

Just like you did before, using the model y = a ₀ + a ₁ x + a ₂ x ², you arrive at the following linear system:

Using the least squares method, you can find a solution for the coefficients a ₀, a ₁, and a ₂ that provides a parabola that minimizes the squared difference between the curve and the data points. For that, you can use the following code:

In this program, you’ve set up the following:

Lines 1 to 2: You import numpy as np and linalg from scipy in order to use linalg.lstsq() .

Lines 4 to 5: You create the coefficients matrix A using a NumPy array called A and the vector with the independent terms b using a NumPy array called b .

Line 7: You calculate the least squares solution for the problem using linalg.lstsq() , which takes the coefficients matrix and the vector with the independent terms as input.

lstsq() provides several pieces of information about the system, including the residues, rank, and singular values of the coefficients matrix. In this case, you’re interested only in the coefficients of the polynomial to solve the problem according to the least squares criteria, which are stored in p .

As you can see, even considering a linear system that has no exact solution, lstsq() provides the coefficients that minimize the squared errors. With the following code, you can visualize the solution provided by plotting the parabola and the data points:

This program uses matplotlib to plot the results:

Line 1: You import matplotlib.pyplot as plt , which is typical.

Lines 3 to 4: You create a NumPy array named x , with values ranging from 0 to 3 , containing 1000 points. You also create a NumPy array named y with the corresponding values of the model.

Line 6: You plot the curve for the parabola obtained with the model given by the points in the arrays x and y .

Lines 7 to 9: In red ( "ro" ), you plot the three points used to build the model.

The output should be the following figure:

Notice how the curve provided by the model tries to approximate the points as well as possible.

Besides lstsq() , there are other ways to calculate least squares solutions using SciPy. One of the alternatives is using a pseudoinverse , which you’ll explore next.

Another way to compute the least squares solution is by using the Moore-Penrose pseudoinverse of a matrix.

You can think of a pseudoinverse as a generalization of the matrix inverse, as it’s equal to the usual matrix inverse when the matrix isn’t singular.

However, when the matrix is singular, which is the case in linear systems that lack a unique solution, then the pseudoinverse computes the matrix that provides the best fit, leading to the least squares solution.

Using the pseudoinverse, you can find the coefficients for the parabola used in the previous example:

This code is very similar to the code from the previous section, except for the highlighted lines:

Line 7: You calculate the pseudoinverse of the coefficients matrix and store it in A_pinv .

Line 9: Following the same approach used to solve linear systems with the inverse of a matrix, you calculate the coefficients of the parabola equation using the pseudoinverse and store them in the vector p2 .

As you’d expect, the least squares solution is the same as the lstsq() solution. In this case, because A is a square matrix, pinv() will provide a square matrix with the same dimensions as A , optimizing for the best fit in the least squares sense:

However, it’s worth noting that you can also calculate pinv() for non-square matrices, which is usually the case in practice. You’ll dive into that next, with an example using real-world data.

Example: Predicting Car Prices With Least Squares

In this example, you’re going to build a model using least squares to predict the price of used cars using the data from the Used Cars Dataset . This dataset is a huge collection with 957 MB of vehicle listings from craigslist.org, including very different types of vehicles.

When working with real data, it’s often necessary to perform some steps of filtering and cleaning in order to use the data to build a model. In this case, it’s necessary to narrow down the types of vehicles that you’ll include, in order to get better results with your model.

Since your main focus here is on using least squares to build the model, you’ll start with a cleaned dataset, which is a small subset from the original one. Before you start working on the code, get the cleaned data CSV file by clicking the link below and navigating to vehicles_cleaned.csv :

In the downloadable materials, you can also check out the Jupyter Notebook to learn more about data preparation.

To load the CSV file and process the data, you’ll use pandas . So, make sure to install it in the conda environment linalg as follows:

After downloading the data and setting up pandas, you can start a new Jupyter Notebook and load the data by running the following code block:

This will create a pandas DataFrame named cars_data containing the data from the CSV file. From this DataFrame, you’ll generate the NumPy arrays that you’ll use as inputs to lstsq() and pinv() to obtain the least squares solution. To learn more on how to use pandas to process data, take a look at Using pandas and Python to Explore Your Dataset .

A DataFrame object includes an attribute named columns that allows you to consult the names of the columns included in the data. That means you can check the columns included in this dataset with the following code:

You can take a look into one of the lines of the DataFrame using .iloc :

As you can see, this dataset includes nine columns, with the following data:

To use this data to build a least squares model, you’ll need to represent the categorical data in a numeric way. In most cases, categorical data is transformed to a set of dummy variables , which are variables that can take a value of 0 or 1 .

As an example of this transformation, consider the column fuel , which can take the value gas or diesel . You could transform this categorical column to a dummy column named fuel_gas that takes the value 1 when fuel is gas and 0 when fuel is diesel .

Note that you’ll need just one dummy column to represent a categorical column that can take two different values. Similarly, for a categorical column that can take N values, you’re going to need N-1 dummy columns, as one of the values will be assumed as the default.

In pandas, you can transform these categorical columns to dummy columns with get_dummies() :

Here, you’re creating a new DataFrame named cars_data_dummies , which includes dummy variables for the columns specified in the columns argument. You can now check the new columns included in this DataFrame:

Now that you’ve transformed the categorical variables to sets of dummy variables, you can use this information to build your model. Basically, the model will include a coefficient for each of these columns—except price , which will be used as the model output. The price will be given by a weighted combination of the other variables, where the weights are given by the model’s coefficients.

However, it’s customary to consider an extra coefficient that represents a constant value that’s added to the weighted combination of the other variables. This coefficient is known as the intercept , and you can include it in your model by adding an extra column to the data, with all the rows equal to 1 :

Now that you have all the data organized, you can generate the NumPy arrays to build your model using scipy.linalg . That’s what you’ll do next.

To generate the NumPy arrays to input in lstsq() or pinv() , you can use .to_numpy() :

The coefficients matrix A is given by all the columns, except price . Vector b , with the independent terms, is given by the values that you want to predict, which is the price column in this case. With A and b set, you can use lstsq() to find the least squares solution for the coefficients:

These are the coefficients that you should use to model price in terms of a weighted combination of the other variables in order to minimize the squared error. As you’ve seen, it’s also possible to get these coefficients by using pinv() with the following code:

One of the nice characteristics of a linear regression model is that it’s fairly easy to interpret. In this case, you can conclude from the coefficients that the value of the car increases approximately $847 as year increases by 1 , which means that the value of the car decreases $847 per year of car age. Similarly, according to the second coefficient, the value of the car decreases approximately $35.39 per 1,000 miles.

Now that you’ve obtained the model, you’ll use it to predict the price of a car.

Using the model given by the least squares solution, you can predict the price for a car represented by a vector with the values for each of the variables used in the model:

So, a 2010 4-cylinder hatchback, with automatic transmission, gas fuel, and 50,000 miles, in good condition, can be represented with the following vector:

You can obtain the prediction of the price by calculating the dot product between the car vector and the vector p of the coefficients. Because both vectors are one-dimensional NumPy arrays, you can use @ to obtain the dot product:

In this example, the predicted price for the hatchback is approximately $6,160. It’s worth noting that the model coefficients include some uncertainty because the data used to obtain the model could be biased toward a particular type of car, for example.

Besides that, the model choice plays a big role in the quality of the estimates. Least squares is one of the most-used techniques to build models because it’s simple and yields explainable models. In this example, you’ve seen how to use scipy.linalg to build such models. For more details on least squares models, take a look at Linear Regression in Python .

Congratulations! You’ve learned how to use some linear algebra concepts with Python to solve problems involving linear models. You’ve discovered that vectors and matrices are useful for representing data and that, by using linear systems, you can model practical problems and solve them in an efficient manner.

In this tutorial, you’ve learned how to:

Linear algebra is a very broad topic. For more information on some other linear algebra applications, check out the following resources:

• Working With Linear Systems in Python With scipy.linalg
• Scientific Python: Using SciPy for Optimization
• Hands-On Linear Programming: Optimization With Python
• NumPy, SciPy, and pandas: Correlation With Python

Keep studying, and feel free to leave any questions or comments below!

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• SciPy v0.18.1 Reference Guide
• Optimization and root finding ( scipy.optimize )

scipy.optimize.linear_sum_assignment ¶

Solve the linear sum assignment problem.

The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a “worker”) and vertex j of the second set (a “job”). The goal is to find a complete assignment of workers to jobs of minimal cost.

Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost

s.t. each row is assignment to at most one column, and each column to at most one row.

This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.

The method used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm.

New in version 0.17.0.

• http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html
• Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly , 2:83-97, 1955.
• Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly , 3: 253-258, 1956.
• Munkres, J. Algorithms for the Assignment and Transportation Problems. J. SIAM , 5(1):32-38, March, 1957.
• https://en.wikipedia.org/wiki/Hungarian_algorithm

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

• First, prefix name of what this variable represents.
• Second is the list of all the variables.
• Third is the lower bound on this variable.
• Fourth variable is the upper bound.
• Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

• Solving Blending Problem in Python using Gurobi
• Transshipment Problem in Python Using PuLP

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• 5. Data Structures
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5. Data Structures ¶

This chapter describes some things you’ve learned about already in more detail, and adds some new things as well.

5.1. More on Lists ¶

The list data type has some more methods. Here are all of the methods of list objects:

Add an item to the end of the list. Equivalent to a[len(a):] = [x] .

Extend the list by appending all the items from the iterable. Equivalent to a[len(a):] = iterable .

Insert an item at a given position. The first argument is the index of the element before which to insert, so a.insert(0, x) inserts at the front of the list, and a.insert(len(a), x) is equivalent to a.append(x) .

Remove the first item from the list whose value is equal to x . It raises a ValueError if there is no such item.

Remove the item at the given position in the list, and return it. If no index is specified, a.pop() removes and returns the last item in the list. It raises an IndexError if the list is empty or the index is outside the list range.

Remove all items from the list. Equivalent to del a[:] .

Return zero-based index in the list of the first item whose value is equal to x . Raises a ValueError if there is no such item.

The optional arguments start and end are interpreted as in the slice notation and are used to limit the search to a particular subsequence of the list. The returned index is computed relative to the beginning of the full sequence rather than the start argument.

Return the number of times x appears in the list.

Sort the items of the list in place (the arguments can be used for sort customization, see sorted() for their explanation).

Reverse the elements of the list in place.

Return a shallow copy of the list. Equivalent to a[:] .

An example that uses most of the list methods:

You might have noticed that methods like insert , remove or sort that only modify the list have no return value printed – they return the default None . [ 1 ] This is a design principle for all mutable data structures in Python.

Another thing you might notice is that not all data can be sorted or compared. For instance, [None, 'hello', 10] doesn’t sort because integers can’t be compared to strings and None can’t be compared to other types. Also, there are some types that don’t have a defined ordering relation. For example, 3+4j < 5+7j isn’t a valid comparison.

5.1.1. Using Lists as Stacks ¶

The list methods make it very easy to use a list as a stack, where the last element added is the first element retrieved (“last-in, first-out”). To add an item to the top of the stack, use append() . To retrieve an item from the top of the stack, use pop() without an explicit index. For example:

5.1.2. Using Lists as Queues ¶

It is also possible to use a list as a queue, where the first element added is the first element retrieved (“first-in, first-out”); however, lists are not efficient for this purpose. While appends and pops from the end of list are fast, doing inserts or pops from the beginning of a list is slow (because all of the other elements have to be shifted by one).

To implement a queue, use collections.deque which was designed to have fast appends and pops from both ends. For example:

5.1.3. List Comprehensions ¶

List comprehensions provide a concise way to create lists. Common applications are to make new lists where each element is the result of some operations applied to each member of another sequence or iterable, or to create a subsequence of those elements that satisfy a certain condition.

For example, assume we want to create a list of squares, like:

Note that this creates (or overwrites) a variable named x that still exists after the loop completes. We can calculate the list of squares without any side effects using:

or, equivalently:

which is more concise and readable.

A list comprehension consists of brackets containing an expression followed by a for clause, then zero or more for or if clauses. The result will be a new list resulting from evaluating the expression in the context of the for and if clauses which follow it. For example, this listcomp combines the elements of two lists if they are not equal:

and it’s equivalent to:

Note how the order of the for and if statements is the same in both these snippets.

If the expression is a tuple (e.g. the (x, y) in the previous example), it must be parenthesized.

List comprehensions can contain complex expressions and nested functions:

5.1.4. Nested List Comprehensions ¶

The initial expression in a list comprehension can be any arbitrary expression, including another list comprehension.

Consider the following example of a 3x4 matrix implemented as a list of 3 lists of length 4:

The following list comprehension will transpose rows and columns:

As we saw in the previous section, the inner list comprehension is evaluated in the context of the for that follows it, so this example is equivalent to:

which, in turn, is the same as:

In the real world, you should prefer built-in functions to complex flow statements. The zip() function would do a great job for this use case:

See Unpacking Argument Lists for details on the asterisk in this line.

5.2. The del statement ¶

There is a way to remove an item from a list given its index instead of its value: the del statement. This differs from the pop() method which returns a value. The del statement can also be used to remove slices from a list or clear the entire list (which we did earlier by assignment of an empty list to the slice). For example:

del can also be used to delete entire variables:

Referencing the name a hereafter is an error (at least until another value is assigned to it). We’ll find other uses for del later.

5.3. Tuples and Sequences ¶

We saw that lists and strings have many common properties, such as indexing and slicing operations. They are two examples of sequence data types (see Sequence Types — list, tuple, range ). Since Python is an evolving language, other sequence data types may be added. There is also another standard sequence data type: the tuple .

A tuple consists of a number of values separated by commas, for instance:

As you see, on output tuples are always enclosed in parentheses, so that nested tuples are interpreted correctly; they may be input with or without surrounding parentheses, although often parentheses are necessary anyway (if the tuple is part of a larger expression). It is not possible to assign to the individual items of a tuple, however it is possible to create tuples which contain mutable objects, such as lists.

Though tuples may seem similar to lists, they are often used in different situations and for different purposes. Tuples are immutable , and usually contain a heterogeneous sequence of elements that are accessed via unpacking (see later in this section) or indexing (or even by attribute in the case of namedtuples ). Lists are mutable , and their elements are usually homogeneous and are accessed by iterating over the list.

A special problem is the construction of tuples containing 0 or 1 items: the syntax has some extra quirks to accommodate these. Empty tuples are constructed by an empty pair of parentheses; a tuple with one item is constructed by following a value with a comma (it is not sufficient to enclose a single value in parentheses). Ugly, but effective. For example:

The statement t = 12345, 54321, 'hello!' is an example of tuple packing : the values 12345 , 54321 and 'hello!' are packed together in a tuple. The reverse operation is also possible:

This is called, appropriately enough, sequence unpacking and works for any sequence on the right-hand side. Sequence unpacking requires that there are as many variables on the left side of the equals sign as there are elements in the sequence. Note that multiple assignment is really just a combination of tuple packing and sequence unpacking.

5.4. Sets ¶

Python also includes a data type for sets . A set is an unordered collection with no duplicate elements. Basic uses include membership testing and eliminating duplicate entries. Set objects also support mathematical operations like union, intersection, difference, and symmetric difference.

Curly braces or the set() function can be used to create sets. Note: to create an empty set you have to use set() , not {} ; the latter creates an empty dictionary, a data structure that we discuss in the next section.

Here is a brief demonstration:

Similarly to list comprehensions , set comprehensions are also supported:

5.5. Dictionaries ¶

Another useful data type built into Python is the dictionary (see Mapping Types — dict ). Dictionaries are sometimes found in other languages as “associative memories” or “associative arrays”. Unlike sequences, which are indexed by a range of numbers, dictionaries are indexed by keys , which can be any immutable type; strings and numbers can always be keys. Tuples can be used as keys if they contain only strings, numbers, or tuples; if a tuple contains any mutable object either directly or indirectly, it cannot be used as a key. You can’t use lists as keys, since lists can be modified in place using index assignments, slice assignments, or methods like append() and extend() .

It is best to think of a dictionary as a set of key: value pairs, with the requirement that the keys are unique (within one dictionary). A pair of braces creates an empty dictionary: {} . Placing a comma-separated list of key:value pairs within the braces adds initial key:value pairs to the dictionary; this is also the way dictionaries are written on output.

The main operations on a dictionary are storing a value with some key and extracting the value given the key. It is also possible to delete a key:value pair with del . If you store using a key that is already in use, the old value associated with that key is forgotten. It is an error to extract a value using a non-existent key.

Performing list(d) on a dictionary returns a list of all the keys used in the dictionary, in insertion order (if you want it sorted, just use sorted(d) instead). To check whether a single key is in the dictionary, use the in keyword.

Here is a small example using a dictionary:

The dict() constructor builds dictionaries directly from sequences of key-value pairs:

In addition, dict comprehensions can be used to create dictionaries from arbitrary key and value expressions:

When the keys are simple strings, it is sometimes easier to specify pairs using keyword arguments:

5.6. Looping Techniques ¶

When looping through dictionaries, the key and corresponding value can be retrieved at the same time using the items() method.

When looping through a sequence, the position index and corresponding value can be retrieved at the same time using the enumerate() function.

To loop over two or more sequences at the same time, the entries can be paired with the zip() function.

To loop over a sequence in reverse, first specify the sequence in a forward direction and then call the reversed() function.

To loop over a sequence in sorted order, use the sorted() function which returns a new sorted list while leaving the source unaltered.

Using set() on a sequence eliminates duplicate elements. The use of sorted() in combination with set() over a sequence is an idiomatic way to loop over unique elements of the sequence in sorted order.

It is sometimes tempting to change a list while you are looping over it; however, it is often simpler and safer to create a new list instead.

5.7. More on Conditions ¶

The conditions used in while and if statements can contain any operators, not just comparisons.

The comparison operators in and not in are membership tests that determine whether a value is in (or not in) a container. The operators is and is not compare whether two objects are really the same object. All comparison operators have the same priority, which is lower than that of all numerical operators.

Comparisons can be chained. For example, a < b == c tests whether a is less than b and moreover b equals c .

Comparisons may be combined using the Boolean operators and and or , and the outcome of a comparison (or of any other Boolean expression) may be negated with not . These have lower priorities than comparison operators; between them, not has the highest priority and or the lowest, so that A and not B or C is equivalent to (A and (not B)) or C . As always, parentheses can be used to express the desired composition.

The Boolean operators and and or are so-called short-circuit operators: their arguments are evaluated from left to right, and evaluation stops as soon as the outcome is determined. For example, if A and C are true but B is false, A and B and C does not evaluate the expression C . When used as a general value and not as a Boolean, the return value of a short-circuit operator is the last evaluated argument.

It is possible to assign the result of a comparison or other Boolean expression to a variable. For example,

Note that in Python, unlike C, assignment inside expressions must be done explicitly with the walrus operator := . This avoids a common class of problems encountered in C programs: typing = in an expression when == was intended.

5.8. Comparing Sequences and Other Types ¶

Sequence objects typically may be compared to other objects with the same sequence type. The comparison uses lexicographical ordering: first the first two items are compared, and if they differ this determines the outcome of the comparison; if they are equal, the next two items are compared, and so on, until either sequence is exhausted. If two items to be compared are themselves sequences of the same type, the lexicographical comparison is carried out recursively. If all items of two sequences compare equal, the sequences are considered equal. If one sequence is an initial sub-sequence of the other, the shorter sequence is the smaller (lesser) one. Lexicographical ordering for strings uses the Unicode code point number to order individual characters. Some examples of comparisons between sequences of the same type:

Note that comparing objects of different types with < or > is legal provided that the objects have appropriate comparison methods. For example, mixed numeric types are compared according to their numeric value, so 0 equals 0.0, etc. Otherwise, rather than providing an arbitrary ordering, the interpreter will raise a TypeError exception.

Table of Contents

• 5.1.1. Using Lists as Stacks
• 5.1.2. Using Lists as Queues
• 5.1.3. List Comprehensions
• 5.1.4. Nested List Comprehensions
• 5.2. The del statement
• 5.3. Tuples and Sequences
• 5.5. Dictionaries
• 5.6. Looping Techniques
• 5.7. More on Conditions
• 5.8. Comparing Sequences and Other Types

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Python Tutorial

File handling, python modules, python numpy, python pandas, python matplotlib, python scipy, machine learning, python mysql, python mongodb, python reference, module reference, python how to, python examples, python arrays.

Note: Python does not have built-in support for Arrays, but Python Lists can be used instead.

Note: This page shows you how to use LISTS as ARRAYS, however, to work with arrays in Python you will have to import a library, like the NumPy library .

Arrays are used to store multiple values in one single variable:

Create an array containing car names:

What is an Array?

An array is a special variable, which can hold more than one value at a time.

If you have a list of items (a list of car names, for example), storing the cars in single variables could look like this:

However, what if you want to loop through the cars and find a specific one? And what if you had not 3 cars, but 300?

The solution is an array!

An array can hold many values under a single name, and you can access the values by referring to an index number.

Access the Elements of an Array

You refer to an array element by referring to the index number .

Get the value of the first array item:

Modify the value of the first array item:

The Length of an Array

Use the len() method to return the length of an array (the number of elements in an array).

Return the number of elements in the cars array:

Note: The length of an array is always one more than the highest array index.

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Looping Array Elements

You can use the for in loop to loop through all the elements of an array.

Print each item in the cars array:

Adding Array Elements

You can use the append() method to add an element to an array.

Add one more element to the cars array:

Removing Array Elements

You can use the pop() method to remove an element from the array.

Delete the second element of the cars array:

You can also use the remove() method to remove an element from the array.

Delete the element that has the value "Volvo":

Note: The list's remove() method only removes the first occurrence of the specified value.

Array Methods

Python has a set of built-in methods that you can use on lists/arrays.

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Python Matrix: Transpose, Multiplication, NumPy Arrays Examples

What is Python Matrix?

• A Python matrix is a specialized two-dimensional rectangular array of data stored in rows and columns. The data in a matrix can be numbers, strings, expressions, symbols, etc. Matrix is one of the important data structures that can be used in mathematical and scientific calculations.

How do Python Matrices work?

The data inside the two-dimensional array in matrix format looks as follows:

Step 1) It shows a 2×2 matrix. It has two rows and 2 columns. The data inside the matrix are numbers. The row1 has values 2,3, and row2 has values 4,5. The columns, i.e., col1, have values 2,4, and col2 has values 3,5.

Step 2) It shows a 2×3 matrix. It has two rows and three columns. The data inside the first row, i.e., row1, has values 2,3,4, and row2 has values 5,6,7. The columns col1 has values 2,5, col2 has values 3,6, and col3 has values 4,7.

So similarly, you can have your data stored inside the nxn matrix in Python. A lot of operations can be done on a matrix-like addition, subtraction, multiplication, etc.

Python does not have a straightforward way to implement a matrix data type.

The python matrix makes use of arrays, and the same can be implemented.

• Create a Python Matrix using the nested list data type

Create Python Matrix using Arrays from Python Numpy package

Create python matrix using a nested list data type.

In Python, the arrays are represented using the list data type. So now will make use of the list to create a python matrix.

We will create a 3×3 matrix, as shown below:

• The matrix has 3 rows and 3 columns.
• The first row in a list format will be as follows: [8,14,-6]
• The second row in a list will be: [12,7,4]
• The third row in a list will be: [-11,3,21]

The matrix inside a list with all the rows and columns is as shown below:

So as per the matrix listed above the list type with matrix data is as follows:

To read data inside Python Matrix using a list.

We will make use of the matrix defined above. The example will read the data, print the matrix, display the last element from each row.

Example: To print the matrix

Example 2: to read the last element from each row, example 3: to print the rows in the matrix, adding matrices using nested list.

We can easily add two given matrices. The matrices here will be in the list form. Let us work on an example that will take care to add the given matrices.

Last will initialize a matrix that will store the result of M1 + M2.

Example: Adding Matrices

To add, the matrices will make use of a for-loop that will loop through both the matrices given.

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Multiplication of Matrices using Nested List

To multiply the matrices, we can use the for-loop on both the matrices as shown in the code below:

• The python library Numpy helps to deal with arrays. Numpy processes an array a little faster in comparison to the list.

To work with Numpy, you need to install it first. Follow the steps given below to install Numpy.

Step 1) The command to install Numpy is :

Step 2) To make use of Numpy in your code, you have to import it.

Step 3) You can also import Numpy using an alias, as shown below:

We are going to make use of array() method from Numpy to create a python matrix.

Example : Array in Numpy to create Python Matrix

Matrix operation using numpy.array().

The matrix operation that can be done is addition, subtraction, multiplication, transpose, reading the rows, columns of a matrix, slicing the matrix, etc. In all the examples, we are going to make use of an array() method.

Matrix Addition

To perform addition on the matrix, we will create two matrices using numpy.array() and add them using the (+) operator.

Matrix Subtraction

To perform subtraction on the matrix, we will create two matrices using numpy.array() and subtract them using the (-) operator.

Matrix Multiplication

First will create two matrices using numpy.arary(). To multiply them will, you can make use of numpy dot() method. Numpy.dot() is the dot product of matrix M1 and M2. Numpy.dot() handles the 2D arrays and perform matrix multiplications.

Matrix Transpose

The transpose of a matrix is calculated, by changing the rows as columns and columns as rows. The transpose() function from Numpy can be used to calculate the transpose of a matrix.

Slicing of a Matrix

Slicing will return you the elements from the matrix based on the start /end index given.

• The syntax for slicing is – [start:end]
• If the start index is not given, it is considered as 0. For example [:5], it means as [0:5].
• If the end is not passed, it will take as the length of the array.
• If the start/end has negative values, it will the slicing will be done from the end of the array.

Before we work on slicing on a matrix, let us first understand how to apply slice on a simple array.

Now let us implement slicing on matrix . To perform slicing on a matrix

the syntax will be M1[row_start:row_end, col_start:col_end]

• The first start/end will be for the row, i.e to select the rows of the matrix.
• The second start/end will be for the column, i.e to select the columns of the matrix.

The matrix M1 tthat we are going to use is as follows:

There are total 4 rows. The index starts from 0 to 3. The 0 th row is the [2,4,6,8,10], 1 st row is [3,6,9,-12,-15] followed by 2 nd and 3 rd .

The matrix M1 has 5 columns. The index starts from 0 to 4.The 0 th column has values [2,3,4,5], 1 st columns have values [4,6,8,-10] followed by 2 nd , 3 rd , 4 th , and 5 th .

Here is an example showing how to get the rows and columns data from the matrix using slicing. In the example, we are printing the 1 st and 2 nd row, and for columns, we want the first, second, and third column. To get that output we have used: M1[1:3, 1:4]

Example : To print all rows and third columns

Example: to print the first row and all columns, example: to print the first three rows and first 2 columns, accessing numpy matrix.

We have seen how slicing works. Taking that into consideration, we will how to get the rows and columns from the matrix.

To print the rows of the matrix

In the example will print the rows of the matrix.

To get the last row, you can make use of the index or -1. For example, the matrix has 3 rows,

so M1[0] will give you the first row,

M1[1] will give you second row

M1[2] or M1[-1] will give you the third row or last row.

To print the columns of the matrix

• Python does not have a straightforward way to implement a matrix data type. Python matrix can be created using a nested list data type and by using the numpy library.
• The matrix operation that can be done is addition, subtraction, multiplication, transpose, reading the rows, columns of a matrix, slicing the matrix, etc.
• To add two matrices, you can make use of numpy.array() and add them using the (+) operator.
• To multiply them will, you can make use of the numpy dot() method. Numpy.dot() is the dot product of matrix M1 and M2. Numpy.dot() handles the 2D arrays and perform matrix multiplications.
• The transpose of a matrix is calculated by changing the rows as columns and columns as rows. The transpose() function from Numpy can be used to calculate the transpose of a matrix.
• Slicing of a matrix will return you the elements based on the start /end index given.