I am studying linear algebra from textbook Hoffman and Kunze and I have a question in a Theorem of Section 9.3 ( Positive Forms) Image of Theorem:
Questions : why in 2nd paragraph g(X, X)$\geq$ 0 holds. Clearly, g(X, X) is a 1×1 matrix but I am unable to understand how it must be non-negative.
(2) How does in next line of above question invertibility of P and X$\neq$ 0 implies ${(PX)^{*} } PX$ >0 ?
If I understand the notation (especially assuming that $P^{*}$ is the conjugate transpose of $P$), $PX$ represents a $n \times 1$ vector, so then $\left(PX\right)^{*}\left(PX\right)$ is the squared magnitude of that vector, and thus must be $\ge 0$ in general.
If $X \ne \mathbf{0}$, then the squared magnitude must be $\gt 0$, in light of the fact that $P$ is invertible, and thus only has $\mathbf{0}$ in its null-space.
I hope this helps.