A counter for: If $f+f^{-1}$ is diagonalizable then $f$ is diagonalizable.

Question

Let $V$ be a finite dimensional vector space and $f:V \rightarrow V$ invertible linear transformation.

Prove or disprove: If $f+f^{-1}$ is diagonalizable then $f$ is diagonalizable.

I know it's not true because I found

$A= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $

for which $A+A^{-1}$ is diagonalizable but $A$ isn't.

First of all, is my assumption about the connection between diagonalization of a matrix and a transformation correct?

If so, how can find an example for such transformation using $A$?

If not, how can I find a counter example?

Thanks!