Schur decomposition

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

$Q^{H}AQ=T=D+N$

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]

References

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

单词	SchurDecomposition
释义	Schur decomposition 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 $Q^{H}AQ=T=D+N$ 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] References GVL Golub, H. Gene, Van Loan F. Charles: Matrix Computations (Third Edition). The Johns Hopkins University Press, London, 1996.
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