My task is the following:
$Let\,\, A,U \in \mathbb{C}^{n\times n} \,\,and\,\,assume \,\,that \,\, U \,\,is \,\,unitary \,\, and \,\, that \,\,U^*AU \,\,is\,\, a\,\, diagonal\,\, matrix.\\Show \,\, that \,\,A \,\,is\,\,normal. $
My approach was:
$M:= U^*AU \Rightarrow A = UMU^*. Now \,\, I \,\, want \,\, to \,\, show \,\, that \,\, A \,\, is \,\, normal \,\,so\,\,I \,\, have \,\, to \,\, show:\\A^*A=AA^*$
But when I now insert $UMU^*$ for A I don't get far. I don't know how to proceed.
Following your notation: \begin{eqnarray} A=UMU^*\Rightarrow A^*&=&UM^*U^*,\\ AA^*=UMU^*UM^*U^*&=&UMM^*U^*,\\ A^*A=UM^*U^*UMU^*&=&UM^*MU^*. \end{eqnarray} Then $A$ is normal, as diagonal matrices $M$ and $M^*$ commute.