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[parent] corollary of Schur decomposition (Corollary)

theorem: $ A\in \mathbb{C}^{n\times n}$ is a normal matrix if and only if there exists a unitary matrix $ Q \in \mathbb{C}^{n\times n}$ such that $ Q^HAQ = \operatorname{diag}(\lambda_1,\ldots , \lambda_n)$(the diagonal matrix) where $ ^H$ is the conjugate transpose. [GVL]

proof: Firstly we show that if there exists a unitary matrix $ Q \in \mathbb{C}^{n\times n}$ such that $ Q^HAQ = \operatorname{diag}(\lambda_1,\ldots , \lambda_n)$ then $ A\in \mathbb{C}^{n\times n}$ is a normal matrix. Let $ D = \operatorname{diag}(\lambda_1,\ldots , \lambda_n)$ then $ A$ may be written as $ A = QDQ^H$. Verifying that A is normal follows by the following observation $ AA^H = QDQ^HQD^HQ^H = QDD^HQ^H$ and $ A^HA = QD^HQ^HQDQ^H = QD^HDQ^H$. Therefore $ A$ is normal matrix because $ DD^H = \operatorname{diag}(\lambda_1\bar{\lambda_1},\ldots , \lambda_n \bar{\lambda_n}) = D^HD$.
Secondly we show that if $ A\in \mathbb{C}^{n\times n}$ is a normal matrix then there exists a unitary matrix $ Q \in \mathbb{C}^{n\times n}$ such that $ Q^HAQ = \operatorname{diag}(\lambda_1,\ldots , \lambda_n)$. By Schur decompostion we know that there exists a $ Q\in \mathbb{C}^{n \times n}$ such that $ Q^HAQ=T$($ T$ is an upper triangular matrix). Since $ A$ is a normal matrix then $ T$ is also a normal matrix. The result that $ T$ is a diagonal matrix comes from showing that a normal upper triangular matrix is diagonal (see theorem for normal triangular matrices).
QED

Bibliography

GVL
Golub, H. Gene, Van Loan F. Charles: Matrix Computations (Third Edition). The Johns Hopkins University Press, London, 1996.



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Cross-references: QED, theorem for normal triangular matrices, diagonal, upper triangular matrix, proof, conjugate transpose, diagonal matrix, unitary matrix, normal matrix
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This is version 4 of corollary of Schur decomposition, born on 2003-06-29, modified 2006-06-21.
Object id is 4413, canonical name is CorollaryOfSchurDecomposition.
Accessed 2312 times total.

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AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
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