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Schur decomposition
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(Theorem)
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If $A$ is a complex square matrix of order n (i.e. $A\in\mathrm{Mat}_n(\mathbb{C})$ , then there exists a unitary matrix $Q \in \mathrm{Mat}_n(\mathbb{C})$ such that
$Q^HAQ = T = D + N$
where $^H$ is the conjugate transpose, $D = \operatorname{diag}(\lambda_1, \dots, \lambda_n)$ (the $\lambda_i$ are eigenvalues of $A$ , and $N \in \mathrm{Mat}_n(\mathbb{C})$ is strictly upper triangular matrix. Furthermore, $Q$ can be chosen such that the eigenvalues $\lambda_i$ appear in any order along the diagonal. [GVL]
- GVL
- Golub, H. Gene, Van Loan F. Charles: Matrix Computations (Third Edition). The Johns Hopkins University Press, London, 1996.
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"Schur decomposition" is owned by Daume.
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Cross-references: diagonal, strictly upper triangular matrix, eigenvalues, conjugate transpose, unitary matrix, order, square matrix, complex
There are 4 references to this entry.
This is version 5 of Schur decomposition, born on 2003-06-19, modified 2006-06-21.
Object id is 4380, canonical name is SchurDecomposition.
Accessed 6252 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) |
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Pending Errata and Addenda
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