# Normal and unitary Matrices

1. Can you help find a $2\times 2$ matrix with eigenvalues $1,-1$ that is not a normal matrix?

I really tried to find one but the matrix I found is also normal!!

2. $A$ hermitian and $B$ unitary matrix and AB=BA I need to show $AB$ is normal matrix.

Well I know that both $A,B$ are normal and that $A=A^*$ and $B=B^{-1}$
Then $(AB)(AB)^* = (AB)(A^*B^*)$ but now Im not sure what is ok to do for showing $AB$ is normal...

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$(AB)^{\ast}=B^{\ast}A^{\ast}$. –  WimC Dec 21 '12 at 19:49

Hint for (1): $A$ has the same eigenvalues as $S A S^{-1}$ for any invertible matrix $S$. Try an $S$ that is not normal.
(2): No, $B^* = B^{-1}$ and $(AB)^* = B^* A^*$. But the statement is not true: if $A$ is hermitian and $B$ is unitary, $AB$ is not normal unless $A^2$ commutes with $B$.
Are you sure there isn't some other assumption, e.g. that $A$ has eigenvalues $1$ and $-1$? –  Robert Israel Dec 21 '12 at 19:52