15-00-SchurDecomposition.tex

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\pmcanonicalname{SchurDecomposition}
\pmcreated{2013-03-22 13:42:12}
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\pmowner{Daume}{40}
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\pmtitle{Schur decomposition}
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\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmrelated{AnExampleForSchurDecomposition}
\pmrelated{ProofThatDetEAEoperatornametrA}

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If $A$ is a complex square matrix of order n \textit{(i.e. $A\in\mathrm{Mat}_n(\mathbb{C})$)}, then there exists a unitary matrix $Q \in \mathrm{Mat}_n(\mathbb{C})$ such that\\
\begin{center}
$Q^HAQ = T = D + N$
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where $^H$ is the conjugate transpose, $D = \operatorname{diag}(\lambda_1,   \dots, \lambda_n)$ \textit{(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.  \cite{1}
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\bibitem[GVL]{1} Golub, H. Gene, Van Loan F. Charles:  Matrix Computations \textit{(Third Edition)}.  The Johns Hopkins University Press, London, 1996.
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