A property of normal matrices

Question

The statement I am trying to understand is that if $A$ is nilpotent and normal, then $A$ is the zero matrix. Here, I believe we take $A$ to be a square matrix over $\mathbb{C}$. Is this related to the fact that the only eigenvalue of $A$ is zero if it is nilpotent?

Answer 1

Any normal matrix is diagonalizable (with a unitary matrix, actually). A nilpotent matrix has only $0$ as its eigenvalue, so a normal nilpotent matrix is similar to the zero matrix.

The zero matrix is obviously only similar to itself (like any scalar matrix).