Numpy | Linear Algebra

The Linear Algebra module of NumPy offers various methods to apply linear algebra on any numpy array. One can find:

• rank, determinant, trace, etc. of an array.
• eigen values of matrices
• matrix and vector products (dot, inner, outer,etc. product), matrix exponentiation
• solve linear or tensor equations and much more!

Example:

import numpy as np A = np.array([[6, 1, 1], [4, -2, 5], [2, 8, 7]]) # Rank of a matrix print("Rank of A:", np.linalg.matrix_rank(A)) # Trace of matrix A print("Trace of A:", np.trace(A)) # Determinant of a matrix print("Determinant of A:", np.linalg.det(A)) # Inverse of matrix A print("Inverse of A:\n", np.linalg.inv(A)) print("Matrix A raised to power 3:\n", np.linalg.matrix_power(A, 3)) 

Output

Rank of A: 3
Trace of A: 11
Determinant of A: -306.0
Inverse of A:
[[ 0.17647059 -0.00326797 -0.02287582]
[ 0.05882353 -0.13071895 0.08496732]
[-0.11764706 0.1503268 0.05228758]]
Matrix A raised to power 3:
[[336 162 228]
[406 162 469]
[698 702 905]]

Matrix and Vector Products

Using dot() function: It returns the dot product of two arrays. It works with both 1D (vectors) and 2D (matrices). When used on 2D arrays, it performs matrix multiplication.

For N dimensions it is a sum product over the last axis of a and the second-to-last of b:

dot(a, b)[i, j, k, m] = sum(a[i, j, :] * b[k, :, m])

import numpy as np # Scalars prod = np.dot(5, 4) print("Dot Product of scalar values:", prod) # 1D array a = 2 + 3j b = 4 + 5j res = np.dot(a, b) print("Dot Product:", res) 

Dot Product of scalar values: 20 Dot Product: (-7+22j) 

Explanation:

a = 2 + 3j
b = 4 + 5j

Now, dot product
= 2(4 + 5j) + 3j(4 - 5j)
= 8 + 10j + 12j - 15
= -7 + 22j

Using vdot() function: This function also returns the dot product, but with one key difference it takes the complex conjugate of the first argument before multiplying. This makes it useful for complex-valued arrays.

import numpy as np # 1D array a = 2 + 3j b = 4 + 5j res = np.vdot(a, b) print("Dot Product:",res) 

Dot Product: (23-2j) 

Explanation:

a = 2 + 3j
b = 4 + 5j

As per method, take conjugate of a i.e. 2 - 3j
Now, dot product = 2(4 - 5j) + 3j(4 - 5j)
= 8 - 10j + 12j + 15
= 23 - 2j

Common Matrix and Vector Product Functions

The following table lists commonly used NumPy functions for performing various types of matrix and vector multiplications:

Solving Equations and Inverting Matrices

Using linal.solve() function: It is used to find the exact solution of a linear system of equations of the form Ax = b, where:

• A is the coefficient matrix
• b is the constant matrix
• x is the unknown we want to solve for

import numpy as np a = np.array([[1, 2], [3, 4]]) b = np.array([8, 18]) print(("Solution of linear equations:", np.linalg.solve(a, b))) 

('Solution of linear equations:', array([2., 3.])) 

Using linalg.lstsq() function: finds the best possible fit minimizing the difference between When a system doesn’t have an exact solution (for example, when it’s overdetermined), predicted and actual values.

import numpy as np import matplotlib.pyplot as plt # x co-ordinates x = np.arange(0, 9) A = np.array([x, np.ones(9)]) # linearly generated sequence y = [19, 20, 20.5, 21.5, 22, 23, 23, 25.5, 24] # obtaining the parameters of regression line w = np.linalg.lstsq(A.T, y)[0] # plotting the line line = w[0]*x + w[1] # regression line plt.plot(x, line, 'r-') plt.plot(x, y, 'o') plt.show() 

Output

Common Functions for Solving and Inverting Matrices

Below are some of the most useful functions to solve, invert or compute pseudoinverses of matrices:

Special Functions in NumPy Linear Algebra

1. Finding the Determinant - numpy.linalg.det(): The determinant is a number that can be calculated from a square matrix. It helps determine whether a matrix is invertible and is often used in solving systems of linear equations.

import numpy as np A = np.array([[6, 1, 1], [4, -2, 5], [2, 8, 7]]) print(("Determinant of A:", np.linalg.det(A))) 

('Determinant of A:', np.float64(-306.0)) 

2. Finding the Trace - numpy.trace(): The trace of a matrix is the sum of its diagonal elements. It’s often used in linear algebra and statistics.

import numpy as np A = np.array([[6, 1, 1], [4, -2, 5], [2, 8, 7]]) print("Trace of A:", np.trace(A)) 

Trace of A: 11 

Common Special Functions

Here’s a list of other specialized linear algebra functions available in NumPy:

Matrix Eigenvalues Functions in NumPy

NumPy provides functions in its linalg (Linear Algebra) module to calculate eigenvalues and eigenvectors of matrices.

Using linalg.eigh() function: It is used for Hermitian (complex symmetric) or real symmetric matrices. It returns two values:

• An array of eigenvalues
• A matrix of eigenvectors (each column corresponds to one eigenvalue)

import numpy as np from numpy import linalg as geek # Creating an array using array function a = np.array([[1, -2j], [2j, 5]]) print("Array is:\n", a) # Calculating eigenvalues and eigenvectors using eigh() function c, d = geek.eigh(a) print("Eigenvalues are:", c) print("Eigenvectors are:\n", d) 

Output

Array is:
[[ 1.+0.j -0.-2.j]
[ 0.+2.j 5.+0.j]]
Eigenvalues are: [0.17157288 5.82842712]
Eigenvectors are:
[[-0.92387953+0.j -0.38268343+0.j ]
[ 0. +0.38268343j 0. -0.92387953j]]

Using linalg.eig() function: It is used for general square matrices (not necessarily symmetric). It also returns both eigenvalues and eigenvectors.

import numpy as np from numpy import linalg as geek # Creating an array using diag function a = np.diag((1, 2, 3)) print("Array is:\n", a) # Calculating eigenvalues and eigenvectors using eig() function c, d = geek.eig(a) print("Eigenvalues are:", c) print("Eigenvectors are:\n", d) 

Array is: [[1 0 0] [0 2 0] [0 0 3]] Eigenvalues are: [1. 2. 3.] Eigenvectors are: [[1. 0. 0.] [0. 1. 0.] [0. 0. 1.]] 

Common Eigenvalue Functions

This table summarizes various functions related to eigenvalue and eigenvector computations: