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

Definition

If A is a square matrix over the complex numbers, then A can be decomposed as

<math>\mathbf{A}= \mathbf{Q} \mathbf{U} \mathbf{Q}^*<math>

where Q is a 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.