# Similarity of matrices in SU(2) [closed]

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.

• Are their diagonal matrix always equal? – Debarghya Kundu Jun 12 '18 at 18:48
• You assumed they are equivalent, so they have the same characteristic equation, determinant, and trace. So they diagonalize to the same matrix. – Cosmas Zachos Jun 12 '18 at 19:10
• Thanks @Cosmas Zachos – Debarghya Kundu Jun 12 '18 at 19:15
• Is Tr(U) = Tr(V) a sufficient condition to prove U and V are similar,given U,V belongs to SU(2) ? – Debarghya Kundu Jun 12 '18 at 19:20
• Actually yes. They have the same characteristic equation, and eigenvalues, so the respective projectors onto these relate them. see answer. – Cosmas Zachos Jun 12 '18 at 19:27

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 $S=R^\dagger T$.