How to find Eigen value and Eigen vector using Python

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     These are the concepts available in Linear algebra. It has the following definitions.

Eigenvalue(λ): It is a scalar value.It shows about the eigen vector stretch and compress in terms of linear transformation.

Eigenvector(v): It is a vector value.It is a non-zero vector which changes the scale alone.

The formula for square matrix A is given below.

Av=λv

Where A represents matrix. ‘v’ denotes eigenvector. ‘λ’ is for eigenvalue.

Let us implement the eigen value and eigen vector in python.

This program imports numpy package with its object np.

A square matrix is created with elements.

Eigenvalues and eigenvectors variables are declared.

There is a function ‘linalg.eig(matrix)’ which generate the eigenvalues and eigenvectors.

Python Program:

import numpy as np

# Declare a square matrix

A = np.array([[5, 3, 2],

              [1,-1, 4],

              [2, 6, 8]])

# Generate eigenvalues and eigenvectors

eigenvalues, eigenvectors = np.linalg.eig(A)

print("Matrix A:\n", A)

print("\nEigenvalues:\n", eigenvalues)

print("\nEigenvectors:\n", eigenvectors)

Output:

Matrix A:

 [[ 5  3  2]

 [ 1 -1  4]

 [ 2  6  8]]

Eigenvalues:

 [11.21110255+0.j  4.        +0.j -3.21110255+0.j]

Eigenvectors:

 [[ 0.4247359 +0.j  0.93638218+0.j  0.21532584+0.j]

 [ 0.31304487+0.j -0.08512565+0.j -0.87645531+0.j]

 [ 0.84947179+0.j -0.34050261+0.j  0.43065168+0.j]]

Let us code create another python code to verify A * v ≈ λ * v:

import numpy as np1

# Square matrix A is created

A = np1.array([[4, 5],

              [3,  8]])

# Generation of eigenvalues and eigenvectors

eigenvalues, eigenvectors = np1.linalg.eig(A)

print("Matrix A:\n", A)

print("\nEigenvalues:\n", eigenvalues)

print("\nEigenvectors (columns):\n", eigenvectors)

# Verification code: A * v ≈ λ * v

for i in range(len(eigenvalues)):

    v = eigenvectors[:, i]      # i-th eigenvector

    λ = eigenvalues[i]          # i-th eigenvalue

    Av = A @ v                # Matrix-vector multiplication

    λv = λ * v

        print(f"\nVerification for eigenvalue {λ:.2f}:")

    print("A @ v =", Av)

    print("λ * v =", λv)

    print("Difference =", Av - λv)

Output:

Matrix A:

 [[4 5]

 [3 8]]

 Eigenvalues:

 [ 1.64110106+0.j 10.35889894+0.j]

 Eigenvectors (columns):

 [[-0.90440309+0.j -0.61810602+0.j]

 [ 0.4266791 +0.j -0.78609474+0.j]]

Verification for eigenvalue 1.64+0.00j:

A @ v = [-1.48421686+0.j  0.70022352+0.j]

λ * v = [-1.48421686+0.j  0.70022352+0.j]

Difference = [4.44089210e-16+0.j 1.11022302e-16+0.j]

Verification for eigenvalue 10.36+0.00j:

A @ v = [-6.4028978 +0.j -8.14307602+0.j]

λ * v = [-6.4028978 +0.j -8.14307602+0.j]

Difference = [-8.8817842e-16+0.j  0.0000000e+00+0.j]

Thus the python code to generate eigenvalue and eigenvector and verification was done successfully. Hope, this code is useful to you. Keep Coding!!!

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