- #1
maNoFchangE
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Homework Statement
Prove that, if ##T,S\in \mathcal{L}(V)## then ##TS## and ##ST## have the same eigenvalues.
Homework Equations
The Attempt at a Solution
Suppose ##T## is written in a basis in which its matrix is upper triangular, and so is ##S## (these bases may be of different list of vectors in ##V##). Since both ##T## and ##S## are upper triangular, ##TS## and ##ST## are also upper triangular. Now, the diagonal element of ##TS## is
$$
\sum_i T_{pi}S_{ip}
$$
which is the same as the diagonal element of ##ST##
$$
\sum_i S_{pi}T_{ip}
$$
Therefore, ##ST## and ##TS## have the same eigenvalues.
What bothers me is my starting assumption; taking ##T## and ##S## in their own upper triangular bases. Since these bases can be a different list of vectors, is it alright then to multiply their matrices?