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.