0
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ Are their diagonal matrix always equal? $\endgroup$ – Debarghya Kundu Jun 12 '18 at 18:48
  • $\begingroup$ You assumed they are equivalent, so they have the same characteristic equation, determinant, and trace. So they diagonalize to the same matrix. $\endgroup$ – Cosmas Zachos Jun 12 '18 at 19:10
  • $\begingroup$ Thanks @Cosmas Zachos $\endgroup$ – Debarghya Kundu Jun 12 '18 at 19:15
  • $\begingroup$ Is Tr(U) = Tr(V) a sufficient condition to prove U and V are similar,given U,V belongs to SU(2) ? $\endgroup$ – Debarghya Kundu Jun 12 '18 at 19:20
  • $\begingroup$ Actually yes. They have the same characteristic equation, and eigenvalues, so the respective projectors onto these relate them. see answer. $\endgroup$ – Cosmas Zachos Jun 12 '18 at 19:27
0
$\begingroup$

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 $S=R^\dagger T$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.