NumPy: Compute the eigenvalues and right eigenvectors of a given square array

NumPy: Linear Algebra Exercise-7 with Solution

Write a NumPy program to compute the eigenvalues and right eigenvectors of a given square array.

Sample Solution :

Python Code :

import numpy as np

# Create a matrix 'm' using np.mat() function
m = np.mat("3 -2; 1 0")

# Display the original matrix 'm'
print("Original matrix:")
print("a\n", m)

# Compute the eigenvalues and eigenvectors of the matrix 'm' using np.linalg.eig() function
w, v = np.linalg.eig(m)

# Display the eigenvalues of the said matrix
print("Eigenvalues of the said matrix", w)

# Display the eigenvectors of the said matrix
print("Eigenvectors of the said matrix", v) 

Sample Output:

Original matrix:
a
 [[ 3 -2]
 [ 1  0]]
Eigenvalues of the said matrix [ 2.  1.]
Eigenvectors of the said matrix [[ 0.89442719  0.70710678]
 [ 0.4472136   0.70710678]]

Explanation:

In the above exercise –

m = np.mat("3 -2;1 0"): This line creates a 2x2 NumPy matrix m with the elements:

[[ 3, -2],

[ 1, 0]]

w, v = np.linalg.eig(m): This line computes the eigenvalues and eigenvectors of the matrix m. The eig function takes a square matrix as input and returns two arrays: one containing the eigenvalues, and the other containing the corresponding eigenvectors as columns.

print( "Eigenvalues of the said matrix", w): This line prints the eigenvalues of the matrix m. In this case, the eigenvalues are 2.0 and 1.0.

print( "Eigenvectors of the said matrix", v): This line prints the eigenvectors of the matrix m as columns of the v array. The eigenvectors corresponding to the eigenvalues 2.0 and 1.0 are [[-0.89442719, 0.70710678], [-0.4472136 , -0.70710678]].

Note:

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by λ (lambda), is the factor by which the eigenvector is scaled.

Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.

Python-Numpy Code Editor:

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