Normal matrix over $\mathbb C$ can be diagonalized to an invertible matrix.

Question

Is it true that

Every normal matrix over $\mathbb C$ can be diagonalized to an invertible matrix.

?

Answer 1

Here is a fact (true over $\mathbb{R}$ or $\mathbb{C}$): if $$ A = U D U^{-1}, $$ and $D$ is invertible, then $A$ is invertible. (Can you prove this?)

Given this fact, you should be able to answer your question.