Let M and N be block diagonal matrices with the same size and partition
If M ~ N , then there exist an invertible matrix S such that MS = SN.
My question is, does S is also block diagonal matrix ?
If yes, and $$ M = Diag(M_1, M_2, ..., M_n) $$ and $$ N = Diag(N_1, N_2, ..., N_n) $$ does it mean that M_i ~ N_i for all i ?