diagonalizable matrix

Question

$A$ is $n\times n$ matrix over $\mathbb C$. must exist that:

1. $A^*A$ is diagonalizable over C
2. $AA^*$ is unitary matrix
3. if $A$ is not diagonalizable over $\mathbb C$ so $AA^*$ is not diagonalizable over $\mathbb C$
4. $i+1$ is not eigenvalue of $A$

I know the answer is 1+4 but I really dont understand why!

• sorry for bad english

Answer 1

1. It's true, as any Hermitian matrix is diagonalizable over $\Bbb C$.
2. It's not necessarily true, for example if $A=2I$, $AA^*=4I$ is not unitary.
3. $AA^*$ is always diagonalizable over $\Bbb C$ (as a Hermitian matrix), independently of the fact that $A$ is diagonalizable or not.
4. May happen ($A=I$ for example) or not (if $A=(1+i)I$).