Eigen decomposition is a powerful method used in linear algebra that allows us to decompose a linear transformation into a unique decomposition into simpler mat...
Eigen decomposition is a powerful method used in linear algebra that allows us to decompose a linear transformation into a unique decomposition into simpler matrices.
Let A be an m x n real matrix, where m and n are positive integers. The eigenvalues of A are the roots of the characteristic polynomial of A, which is a polynomial of degree n that can be expressed in the form:
where **(\lambda_1, \lambda_2, ..., \lambda_r) are the eigenvalues of A and k_1, k_2, ..., k_r are the corresponding multiplicities of each eigenvalue.
The eigenvectors of A are the columns of A that correspond to the distinct eigenvalues.
The eigen decomposition of A is given by the following formula:
where:
U is an m x n orthogonal matrix containing the eigenvectors of A in the columns.
Σ is an m x n diagonal matrix containing the eigenvalues of A on the diagonal.
V^T is the transpose of an n x m matrix containing the eigenvectors of A in the columns.
The eigen decomposition has several important properties, including:
U is a unitary matrix, meaning its columns are orthogonal unit vectors.
Σ is a diagonal matrix, where the diagonal elements are the eigenvalues of A.
V^T is the transpose of an orthogonal matrix, which means its columns are orthogonal unit vectors.
The eigen decomposition can be used to decompose any linear transformation. It can also be used to solve systems of linear equations and to analyze the behavior of linear transformations