Schur decomposition

If $A$ is a complex square matrix of order n (i.e. $A\mathrm{\in }{\mathrm{Mat}}_n\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$), then there exists a unitary matrix $Q\in {\mathrm{Mat}}_n(ℂ)$ such that

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

where ${}^H$ is the conjugate transpose, $D=\mathrm{diag}({\lambda }_1,\mathrm{\dots },{\lambda }_n)$ (the ${\lambda }_i$ are eigenvalues of $A$), and $N\in {\mathrm{Mat}}_n(ℂ)$ is strictly upper triangular matrix. Furthermore, $Q$ can be chosen such that the eigenvalues ${\lambda }_i$ appear in any order along the diagonal. [GVL]

Title	Schur decomposition
Canonical name	SchurDecomposition
Date of creation	2013-03-22 13:42:12
Last modified on	2013-03-22 13:42:12
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	8
Author	Daume (40)
Entry type	Theorem
Classification	msc 15-00
Related topic	AnExampleForSchurDecomposition
Related topic	ProofThatDetEAEoperatornametrA