This is a very common theorem. See here, for example.
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.
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 ?
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
Thanks for the suggesting the site.
I already seen that proof and I find it a bit wordy...
I'm thinking why to involve scalar matrices and why the need to induct on vector space dimension... (I understand that this might be also a way to prove)
Eventually I find a very nice and clear proof here: on page 8.