commuting matrices are simultaneously triangularizable

Theorem 1.

All matrices in the below are complex $n\mathrm{\times }n$ matrices.
Let $A$,$B$ be matrices and $A\mathit{}B\mathrm{=}B\mathit{}A$. Then there exists a unitary matrix $Q$ such that

$Q^{H}AQ=T_1$ , $Q^H\mathit{}B\mathit{}Q\mathrm{=}{T}_{\mathrm{2}}$

where ${}^H$ is the conjugate transpose and $T_{\mathrm{1}}\mathrm{,}{T}_{\mathrm{2}}\mathrm{,}$ are upper triangular matrices.

Title	commuting matrices are simultaneously triangularizable
Canonical name	CommutingMatricesAreSimultaneouslyTriangularizable
Date of creation	2013-03-22 15:26:48
Last modified on	2013-03-22 15:26:48
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	12
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 15A23
Related topic	SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices