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Schur decomposition

In mathematics, the Schur decomposition of matrix theory, or Schur triangulation, is an important matrix decomposition

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


where Σ is an triangular matrix whose diagonal entries are exactly the eigenvalues of A. This is the Schur decomposition of A.

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.