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: