# 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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