.eig()

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DanielCrigan
Published Apr 25, 2025
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The .eig() function in the numpy.linalg module computes the eigenvalues and right eigenvectors of a square matrix. Eigendecomposition is a fundamental matrix factorization method that breaks down a matrix into its constituent eigenvalues and eigenvectors, revealing important properties about the linear transformation represented by the matrix.

Eigenvalues and eigenvectors are essential in various applications, including Principal Component Analysis (PCA), solving differential equations, quantum mechanics, vibration analysis, and stability analysis. They provide insights about a matrix’s behavior, such as its scaling properties, diagonalizability, and characteristic dynamics.

Syntax

numpy.linalg.eig(a)

Parameters:

  • a: A square matrix of shape (M, M) used as input to compute its eigenvalues and corresponding right eigenvectors.

Return value:

  • w: An ndarray of shape (M,), containing the eigenvalues of the matrix.
  • v: An ndarray of shape (M, M), containing the eigenvectors of the matrix, where each column v[:, i] corresponds to the eigenvalue w[i].

Notes:

  • .eig() computes right eigenvectors. To obtain left eigenvectors, compute the right eigenvectors of the transposed matrix: v_left = np.linalg.eig(a.T)[1].
  • Eigenvalues may be complex even for real matrices.
  • The eigenvectors are normalized to have a unit length.

Example 1: Basic Eigenvalue Decomposition

This example demonstrates computing eigenvalues and eigenvectors of a simple square matrix:

import numpy as np


# Create a 2x2 matrix
A = np.array([[4, 2],
              [1, 3]])
print("Original matrix:")
print(A)


# Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)


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


print("\nEigenvectors (column-wise):")
print(eigenvectors)


# Verify the eigendecomposition: A * v = lambda * v
# For the first eigenvalue-eigenvector pair
print("\nVerification for first eigenvector:")
print("A * v:", np.dot(A, eigenvectors[:, 0]))
print("lambda * v:", eigenvalues[0] * eigenvectors[:, 0])
print(
  "Are they equal?",
  np.allclose(np.dot(A, eigenvectors[:, 0]), eigenvalues[0] * eigenvectors[:, 0])
)

The output for the example will be:

Original matrix:
[[4 2]
 [1 3]]


Eigenvalues:
[5. 2.]


Eigenvectors (column-wise):
[[0.89442719 0.70710678]
 [0.4472136  0.70710678]]


Verification for first eigenvector:
A * v: [4.4721359 2.236068 ]
lambda * v: [4.4721359 2.236068 ]
Are they equal? True

The code creates a 2×2 matrix, calculates its eigenvalues and eigenvectors, and verifies that the eigendecomposition satisfies the equation A⋅v = λ⋅v, where A is the matrix, v is an eigenvector, and λ is the corresponding eigenvalue.

Example 2: Complex Eigenvalues

This example shows how eigenvalues can be complex for certain matrices:

import numpy as np


# Create a matrix with complex eigenvalues
B = np.array([[0, -1],
              [1, 0]])
print("Matrix B:")
print(B)


# Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(B)


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


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


# Verify the eigendecomposition for complex eigenvalues
print("\nVerification for first eigenvector:")
print("B * v:", np.dot(B, eigenvectors[:, 0]))
print("lambda * v:", eigenvalues[0] * eigenvectors[:, 0])
print(
  "Are they equal?",
  np.allclose(np.dot(B, eigenvectors[:, 0]), eigenvalues[0] * eigenvectors[:, 0])
)

The output for the example will be:

Matrix B:
[[ 0 -1]
 [ 1  0]]


Eigenvalues:
[0.+1.j 0.-1.j]


Eigenvectors:
[[0.70710678+0.j         0.70710678-0.j        ]
 [0.        +0.70710678j 0.        -0.70710678j]]


Verification for first eigenvector:
B * v: [0.        -0.70710678j 0.70710678+0.j        ]
lambda * v: [0.        +0.70710678j 0.70710678+0.j        ]
Are they equal? True

This example demonstrates how the .eig() function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues 1j and -1j, illustrating how complex eigenvalues often indicate rotational transformations.

Codebyte Example: Eigendecomposition of a Symmetric Matrix

This example explores the special properties of eigendecomposition for symmetric matrices:

Code
Output
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