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Diagonal Complex Matrix Characteristic Values And Vectors Of A Matrix Square Matrix Unitary Transformation Conjugate Transpose Strict Upper Triangular Matrix

1	Diagonal
1	Matrix
1	Complex
2	Characteristic Values And Vectors Of A Matrix
2	Square Matrix
4	Unitary Transformation
5	Conjugate Transpose
13	Strict Upper Triangular Matrix

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Schur Decomposition

If 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

where

is the conjugate transpose, $D = \operatorname{diag}(\lambda_1, \dots, \lambda_n)$ (the $\lambda_i$ are eigenvalues of

), and $N \in \mathrm{Mat}_n(\mathbb{C})$ is strictly upper triangular matrix. Furthermore,

can be chosen such that the eigenvalues $\lambda_i$ appear in any order along the diagonal. [GVL]

Bibliography

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