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 $W$.