Spectral decomposition and eigendecomposition are both fundamental concepts in linear algebra, used to break down matrices into simpler components. While they are closely related, they differ in their application scope and conditions.
Eigen Decomposition
Eigen Decomposition applies to square matrices (i.e., matrices with the same number of rows and columns). It decomposes a matrix 𝐴 into the form:
A = PDP^{-1}A is a matrix whose columns are the eigenvectors of A, and D is a diagonal matrix with the eigenvalues of A on its diagonal. Eigen Decomposition is possible when the matrix P(the matrix of eigenvectors) is invertible, meaning the eigenvectors are linearly independent.
Spectral Decomposition
Spectral Decomposition specifically applies to symmetric (or Hermitian) matrices. It’s a special case of Eigen Decomposition and can be expressed as:
H = Q\Lambda Q^T
In this equation, Q is an orthogonal matrix whose columns are the eigenvectors of A. Λ is a diagonal matrix, and its diagonal elements are the eigenvalues. Q^T is the transpose of Q for symmetric matrices, or the conjugate transpose for Hermitian matrices. The key feature of spectral decomposition is that Q is an orthogonal matrix.
Difference
- Applicable Matrix Types: Eigendecomposition can be applied to any square matrix, whereas spectral decomposition is limited to symmetric or Hermitian matrices.
- Properties of Matrices ( P ) and ( Q ): In eigendecomposition, ( P ) just needs to be invertible, but in spectral decomposition, ( Q ) must be an orthogonal matrix.
Reference
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