Eigenvalue Decomposition
Eigenvalue decomposition factors a square matrix A as A=QΛQ^(-1), with Λ the diagonal matrix of eigenvalues and Q the eigenvector matrix, revealing the matrix's action.
Purpose
To analyze the essence of a matrix transformation (scaling and rotation), used in stability analysis and spectral clustering.
Mechanism
Solve det(A-λI)=0 for the eigenvalues λ_i, then (A-λ_i I)v=0 for the eigenvectors. Symmetric matrices have real eigenvalues and orthogonal eigenvectors. Non-symmetric ones may have complex eigenvalues and require Schur decomposition.
Use cases
- Spectral clustering
- Principal component analysis
- Dynamical system analysis