If $B = UD{U^*}$ is real matrix and $D$ is diagonal matrix whit ${d_i} \ne {d_j}$. then $U=VW$ Let $B\in M_n$ * $B = UD{U^*}$ is real matrix and $D$ is diagonal matrix whit $d_i$ is real positive and ${d_i} \ne {d_j}$.for $i,j=1,2, ...., n$. * $U$ is unitary matrix Can we prove that $U=VW$ where $V$ is real orthogonal and $W$ is a diagonal unitary matrix?

The answer to this one is yes!

In particular, note by the real spectral theorem that there is a (real) orthogonal $V$ such that $$ UDU^* = B = VDV^* $$ Now, we have $$ UDU^* = VDV^* \implies\\\ (V^*U)D(V^*U)^* = D \implies\\\ (V^*U)D = D(V^*U) $$ Which is to say that $V^*U$ commutes with $D$. _Verify_ that this can only occur if $V^*U$ is diagonal (of particular importance is the fact that the $d_i$ are distinct).

So, setting $W = V^*U$, we have $U = VW$, as desired.