Discussion
Let \(A \in M_{n}(\mathbb{C})\) be a normal matrix. Consider the following statements.
I. If all the eigenvalues of \(A\) are real, then \(A\) is Hermitian.
II. If all the eigenvalues of \(A\) have absolute value 1, then \(A\) is unitary.
Which one of the following is correct?
Answer: A
Both I and II are TRUE.
A normal matrix is unitarily diagonalisable: \(A = UDU^{*}\) with \(U\) unitary and \(D\) diagonal holding the eigenvalues.
I: if the eigenvalues are real, \(D^{*} = D\), so \(A^{*} = UD^{*}U^{*} = A\) and \(A\) is Hermitian. II: if every eigenvalue has modulus 1, then \(D^{*}D = I\), so \(A^{*}A = UD^{*}DU^{*} = I\) and \(A\) is unitary.
Normality is essential: without it, neither statement holds (a non-normal upper triangular matrix with real diagonal is not Hermitian). With it, both are true, so the other three options are wrong.
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