Eigendecomposition

Eigen decomposition is a method of decomposing a matrix into the product of matrices represented by eigenvalues and eigenvectors, but only diagonalizable matrices can perform eigendecomposition.

The eigenvalue can be regarded as the scaling ratio of the length of the eigenvector under linear change. If the eigenvalue is positive, it means that the direction of $latex v $ remains unchanged after the linear transformation; if the eigenvalue is negative, it means that the direction will be reversed; if the eigenvalue is 0, it means that it shrinks to zero.

Eigendecomposition of a standard matrix

Assume that A is a square matrix of N x N with N linearly independent eigenvectors Qi (i = 1, 2, 3, ····, N), where A can be decomposed into $latex \mathbf{A}=\mathbf{Q} \mathbf{\Lambda} \mathbf{Q}^{-1} $

Where Q is an N x N square matrix with the i-th column being the eigenvector Qi of A, and Λ is a diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e. $latex \Lambda_{ii}=\lambda_{i} $

Eigendecomposition of a symmetric matrix

Any N x N real symmetric matrix has N linearly independent eigenvectors, and they can all be orthogonally normalized to obtain a set of orthogonal vectors with modulus 1, so the symmetric matrix A can be decomposed into $latex \mathbf{A}=\mathbf{Q} \mathbf{\Lambda} \mathbf{Q}^{T} $

Eigendecomposition of a normal matrix

Similarly, a complex normal matrix has a set of orthogonal eigenvector basis, so the normal matrix can be decomposed into $latex \mathbf{A}=\mathbf{U} \mathbf{\Lambda} \mathbf{U}^{H} $

Where U is a unitary matrix, we can conclude that if A is a Hermitian matrix, then the diagonal elements of the diagonal matrix Λ are all real numbers; if A is a unitary matrix, then all diagonal elements of Λ are taken on the unit circle in the complex plane.