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
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.it:Decomposizione di Schur