Overview
An eigenvector of a matrix is a column vector that when left multiplied by the matrix, returns the same column vector, possibly multiplited by a scaler. The column vector is referred to as an eigenvector, and the corresponding scale is called the eigenvalue.
{% A\vec{v} = a\vec{v} %}
let la = await import('/lib/linear-algebra/v1.0.0/linear-algebra.mjs');
let matrix1 = [[1,2],[3,4]];
let ans = la.eigenvalues(matrix1);
The following demonstrates calcuating the eigenvalues/vectors in Python
import numpy as np
# Define a square matrix
A = np.array([[4, 2],
[1, 3]])
# Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)
print("Eigenvalues:\n", eigenvalues)
print("Eigenvectors (columns):\n", eigenvectors)
Eigen Decomposition
{% A = Q \Sigma Q^{-1} %}
Q is an nxn matrix whose columns are eigenvectors of A and {% \Sigma %} is a diagonal matrix of eigenvalues. The eigendecomposition
does not exist for all matrices. However, a similar decomposition, the
SVD decomposition, does exist for all matrices.
see spectral decomposition
Topics
- Symmetric Matrices {% A^TA %} and {% AA^T %}
- Raleigh Quotient
- QR Algorithm