I think we are supposed to use Shurs Theorem of diagnolizabl

I think we are supposed to use Shur\'s Theorem of diagnolizable matrices but not completely sure.

2. Let M be an n x n matrix with entries in C such that M*M MM. and M M2. (M* is the conjugate transpose of M) (a) Prove M2 M (b) Prove or give a counterexample: M I

Solution

Given M^* M= M M^*=I (Unitry matrix property)

M³=M²

Multiplying both sides by M^*

M³ M*= M²M^*

M² (M M^*) = M(M M*)

Thus, M²= M

Further multiplying by M * we can prove that M=I