Help to prove numerically the given equation below?

Question

Consider a spectral decomposition of a unitary matrix $U$ given by $WAW^*$ where $A$ is diagonal matrix of eigen-values of $U$ and the symbol $^*$ means transconjugate. An infinitesimal shift $dU$would change the matrices by $dA$ and $dW$. Can anyone please help me to prove numerically that $WdW^*+dWW^*=0$. Any comments would be highly appreciated. I can see how Hermitian property can lead to the result but I need help to show numerically. Specifically, is there a way to compute $dW$ without any model of $W$.

Answer 1

Since $U$ is normal, you can choose $W$ as a unitary matrix s.t. $U=WAW^*$ and $A$ is diagonal complex. Since $WW^*=I$, we deduce that $WdW^*+dWW^*=0$ (with the transconjugate $W^*$ and not with the transpose $W'$ !).