Given a *n*-by-*n* square matrix *A*, there exists a unitary matrix *U* such that

*A*=*UΣU*^{*}

If *A* is a normal matrix, then Σ is a diagonal matrix and the column vectors of *U* are the eigenvectors of *A* and the schur decomposition is called the spectral decomposition. Furthermore, if *A* is positive definite, the Schur decomposition of *A* is the same as the singular value decomposition of the matrix.