Eigenvalues are special numbers associated with a matrix that provide important information about the matrix’s properties.
In the context of linear algebra, if A is a square matrix, an eigenvalue is a scalar λ such that there exists a non-zero vector v (called an eigenvector) satisfying the equation −
Av = λv
This means that when the matrix A multiplies the vector v, the result is the same as multiplying the vector v by the scalar λ.
NumPy provides the numpy.linalg.eig() function to compute the eigenvalues and eigenvectors of a square matrix. Let us see how this function works with an example.
In this example, the eigenvalues of the matrix A are 3 and 2. The corresponding eigenvectors are shown in the output −
# Open Compiler
import numpy as np
# Define a 2x2 matrix
A = np.array([[4, -2],
[1, 1]])
# Compute the eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)
Output:
The output from numpy.linalg.eig() function provides two arrays: one for eigenvalues and one for eigenvectors.
The eigenvalues array contains the eigenvalues of the matrix, and each column of the eigenvectors array represents an eigenvector corresponding to the respective eigenvalue −
Eigenvalues: [3. 2.]
Eigenvectors:
[[ 0.89442719 0.70710678]
[ 0.4472136 -0.70710678]]
Eigenvalues and eigenvectors have several important properties. They are −
Eigenvalues and eigenvectors have numerous applications, such as −
In the following example, we are computing the eigenvalues and eigenvectors of a 3×3 matrix using NumPy −
# Open Compiler
import numpy as np
# Define a 3x3 matrix
B = np.array([[1, 2, 3],
[0, 1, 4],
[5, 6, 0]])
# Compute the eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(B)
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)
Output:
This will produce the following result −
Eigenvalues: [-5.2296696 -0.02635282 7.25602242]
Eigenvectors:
[[ 0.22578016 -0.75769839 -0.49927017]
[ 0.52634845 0.63212771 -0.46674201]
[-0.81974424 -0.16219652 -0.72998712]]
A symmetric matrix is a matrix that is equal to its transpose (i.e., A = AT). Symmetric matrices have some special properties regarding their eigenvalues −
Let us compute the eigenvalues of a symmetric matrix −
# Open Compiler
import numpy as np
# Define a symmetric matrix
C = np.array([[4, 1, 1],
[1, 4, 1],
[1, 1, 4]])
# Compute the eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(C)
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)
Output:
Following is the output of the above code −
Eigenvalues: [6. 3. 3.]
Eigenvectors:
[[-0.57735027 -0.81649658 -0.15430335]
[-0.57735027 0.40824829 -0.6172134 ]
[-0.57735027 0.40824829 0.77151675]]
A square matrix A is said to be diagonalizable if it can be written as −
A = PDP-1
where, D is a diagonal matrix containing the eigenvalues of A, and P is a matrix whose columns are the eigenvectors of A.
Let us see how to diagonalize a matrix using NumPy −
# Open Compiler
import numpy as np
# Define a matrix
D = np.array([[2, 0, 0],
[1, 3, 0],
[4, 5, 6]])
# Compute the eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(D)
# Diagonal matrix of eigenvalues
D_diag = np.diag(eigenvalues)
# Reconstruct the original matrix
reconstructed_D = eigenvectors @ D_diag @ np.linalg.inv(eigenvectors)
print("Original matrix:\n", D)
print("Reconstructed matrix:\n", reconstructed_D)
Output:
The original matrix is successfully reconstructed using its eigenvalues and eigenvectors, demonstrating the process of diagonalization −
Original matrix:
[[2 0 0]
[1 3 0]
[4 5 6]]
Reconstructed matrix:
[[2. 0. 0.]
[1. 3. 0.]
[4. 5. 6.]]
Key Takeaway: Master eigenvalues and eigenvectors with NumPy at Vista Academy!