Existence of the matrix Z

Question

Given a non-diagonalizable matrix $A$, find the matrix $Z$ so that $ZAZ^{-1}$ is a Hermitian matrix.

Answer 1

If $ ZAZ^{-1}$ is hermitian, then $ ZAZ^{-1}$ is diagonalizable.

Hence, with an invertble matrix $X$, we have $X(ZAZ^{-1})X^{-1}=D$, where $D$ is diagonal.

It follows:

$(XZ)A(XZ)^{-1}=D$

and $A$ is diagonalizable.

Conclusion: a matrix $Z$ with the desired properties doe not exist.