hello,

i am having some trouble understanding simultaneous diagonalization. i have understood the proof which tells us that two hermitian matrices can be simultaneously diagonalized by the same basis vectors if the two matrices commute. but my book then shows a proof for the case when the matrices are both degenerate. it is basically going over my head, will anyone please help me with a good proof/ sketch?

my book uses block diagonal elements for the proof which i cannot understand....

thank you.