Schur decomposition

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In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important matrix decomposition.

Definition

Let A be a square matrix over a field, then A can be decomposed as

where Q is an unitary matrix, Q^* is the conjugate transpose of Q and U is an upper triangular matrix whose diagonal entries are exactly the eigenvalues of A.

Notes

If A is a normal matrix, then U is even a diagonal matrix and the column vectors of Q 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.