Prove that if $T$ and $U$ are simultaneously diagonalizable linear operators on a finite-dimensional vector space $V$, then the matrices $[T]_\beta$ and $[U]_\beta$ are simultaneously diagonalizable for any ordered basis $\beta$.
Does that mean I only need to find an invertible matrix $Q$ such that $Q^{-1} [T]_\beta Q$ and $Q^{-1} [U]_\beta Q$ are diagonalizable? I am totally confused!
Can I prove the "there exists" statement by just finding an invertible identity matrix such that $I^{-1} [T]_\beta I$ and $I^{-1} [U]_\beta I$ are diagonalizable? thanks for helping.