A special case of singular value decomposition is eigenvalue decomposition: a symmetric matrix can be decomposed into

such that is diagonal and is an orthogonal matrix (and thus ).

It also happens that ‘s diagonal contains the eigenvalues of and the columns of are eigenvectors.

Eigenvalue Decomposition of Asymmetric Matrix

If we allow eigenvalues and eigenvectors to be complex (numbers), then we can do eigenvalue decomposition for asymmetric matrix.