Linear transformations and eigenvalues

Question

Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same.

Can anyone provide tips/hints in the right direction? I'm struggling as I try to understand this intuitively....thank you

Answer 1

Hint:

$[T]_\beta$ and $[T]_\gamma$ are similar matrices, i.e. there is a matrix $Q$ (change of coordinate matrix) so that $[T]_\beta = Q^{-1}[T]_\gamma Q$.

Answer 2

The eigenvalues are a property of the operator $T$. Recall that $\lambda$ is an eigenvalue of $[T] $ if $0=\det([T]-\lambda\,I)=\det([T-\lambda\,I]) $. This happens precisely when the operator $T-\lambda\, I$ is not invertible, i.e. there exists $v\in\mathbb C$ with $Tv=\lambda v$. This shows that the eigenvalues of $[T]$ (in any basis) are precisely the eigenvalues of $T$.

Answer 3

HINT $$det(B - \lambda I_n) = det(A - \lambda I_n)$$ for two similar matrices $A$ and $B$. (Why?)