91Ó°ÊÓ

Given \(A \in \mathbb{C}^{m \times n}\) with \(m \geq n\), show that \(A^{*} A\) is nonsingular if and only if \(A\) has full rank.

Let \(P \in \mathbb{C}^{m \times m}\) be a nonzero projector. Show that \(\|P\|_{2} \geq 1\), with equality if and only if \(P\) is an orthogonal projector.