When is a matrix similar to the companion matrix of its characteristic polynomial?

Question

Let $A$ be a complex matrix and $A_c$ the companion matrix of its characteristic polynomial. From what I have read, I believe the following two statements to be true: (1) not every $A$ is similar to $A_c$, and (2) $A$ is similar to $A_c$ if $A$ has a simple spectrum. My question would be then, is $A$ similar to $A_c$ if and only if the spectrum of $A$ is simple? or, Are there other conditions under which we can have $A \sim A_c$? I am particularly interested in complex Hermitian matrices.

Answer 1

(For sake of having an answer, let's turn the comments to this question into one.)

The minimal polynomial and characteristic polynomial of a companion matrix are equal. So, if $A\sim A_c$, the minimal and characteristic polynomials of $A$ are equal, too. In other words, in the Jordan form of $A$, every eigenvalue is associated with only one Jordan block. Alternatively speaking, the geometric multiplicity of each eigenvalue is equal to $1$.

When $A$ is Hermitian, geometric multiplicities and algebraic multiplicities coincide. Hence the above necessary and sufficient condition reduces to that all eigenvalues of $A$ are simple, i.e. $A$ has distinct eigenvalues.