## Characterization of a normal matrix

A matrix with complex entries is called normal if , where is the conjugate transpose of . Prove that is normal if and only if .

**Hint: **Prove that if is a matrix of rank then .

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Categories: Higher Algebra, Olympiad
adjoint, matrix

What happens when rank ?

When rank of is use the hint, since the trace of this matrix is zero and therefore . Then if you denote and note that you get , and therefore .