Complex circulant matrices Can someone please show my why a matrix is normal if it is made diagonal by a unitary similarity transformation?
I know that a complex matrix is normal when it commutes with its conjugate transpose, but I am lost on how to do this problem. I suspect I need to do some sort of decomposition by finding the eigenvalues and eigenvectors, but I am not quite sure because I am having trouble finding online material on similarity transformations.

Suppose that $A=UDU^*$. Then $A^*=U^{**}D^*U^* = UD^*U^*$, and since diagonal matrices always commute we have
$$ AA^* = UDU^*UD^*U^* = UDD^*U^* = UD^*DU^* = UD^*U^*UDU^* = A^*A $$
