Two n × n matrices A, B are said to be simultaneously diagonalizable if there is a nonsingular matrix S such that both S-1 A S and S-1 B S are diagonal matrices.
(a) Show that simultaneously diagonalizable matrices commute: AB = BA.
(b) Prove that the converse is valid, provided that one of the matrices has no multiple eigenvalues.
(c) Is every pair of commuting matrices simultaneously diagonalizable?