Linear Algebra: Eigenvalues

Definition

An eigenvalue $\lambda$ of a matrix $A$ is a scalar such that

A\mathbf{v} = \lambda \mathbf{v}

for some nonzero vector $\mathbf{v}$ , called the eigenvector.

Finding Eigenvalues

We solve the characteristic equation:

\det(A - \lambda I) = 0

Example

Let $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ .

\det\begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix} = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10 = 0

So $\lambda_1 = 5$ , $\lambda_2 = 2$ .

Computing with NumPy

import numpy as np

A = np.array([[4, 1],
              [2, 3]])

eigenvalues, eigenvectors = np.linalg.eig(A)
print(f"Eigenvalues: {eigenvalues}")
print(f"Eigenvectors:\n{eigenvectors}")

Eigenvalues: [5. 2.]
Eigenvectors:
[[ 0.70710678 -0.4472136 ]
 [ 0.70710678  0.89442719]]

Key Properties

The trace of $A$ equals the sum of eigenvalues: $\text{tr}(A) = \sum \lambda_i$
The determinant equals the product: $\det(A) = \prod \lambda_i$
A matrix is invertible iff no eigenvalue is zero
Symmetric matrices have real eigenvalues and orthogonal eigenvectors

Flashcards

What is the characteristic equation of a matrix

A

\det(A - \lambda I) = 0

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Quiz

Test Your Knowledge

Question 1 of 4

What are the eigenvalues of

I_n

(the

n \times n

identity matrix)?