Similarity of matrices in SU(2)

Question

I have two matrices U,V in SU(2) such that U = SVS* for some matrix S in SU(2) and S* denote conjugate-transpose of the matrix.What is the easiest way to find S given V and U.

Answer 1

Diagonalize $U$, so $TUT^\dagger=D$, for some unitary $T$.

It then follows that $$ V=S^\dagger U S=S^\dagger T^\dagger D TS, $$ so $$ TSVS^\dagger T^\dagger =D \equiv RVR^\dagger,$$ likewise.

So, diagonalizing U with T and V with R and equating their diagonal forms D yields $S=R^\dagger T$.