Hi,

I'm training for exam and come across a question from one of the exams:

One way to prove this is to find a matrix U such that:T and S are normal linear transformations. Also they commute TS = ST.

It is required to prove that there exists common vector basis consisting of T's (or S's) eigenvectors.

$\displaystyle U^{-1}[T]U = diag(\gamma_i) \mbox{ and } U^{-1}[S]U = diag(\lambda_i)$

After some manipulation I can show that TS and ST are also normal, not sure if this is useful.

I know (i searched the web to find more clues) that commute matrices should have common eigenvector, though again i didn't find simple proof for this fact.

Can anyone please give some direction of how to prove this ?

Thank you

P.S.

Our material doesn't include things like classification into groups/subgroups or representation theory so proof should use more elementary concepts like eigenvector, eigenvalues, unitary matrices