Eigenvalues of a Hermitian matrix and a Herminitian form

Question

Need some help and hints on how to prove this one:

Let $F=\mathbb{R}$ or $\mathbb{C}$, and $_FV=M_{n,1}(F)$. Let $A \in M_n(F)$ be Hermitian (i.e $A^* = \bar{A}^T=A$) and $f(x,y)=x^*Ay$, for all $x,y \in V$. Show that $f$ is a Hermitian form, and that $f$ is an inner product on $_FV$ if and only if all the eigenvalues of $A$ are positive.

I already proved that $f$ is a Hermitian form.

Can I have some help on the if and only if part? Thanks a lot.

Answer 1

Hint: suppose that $f$ is an inner product. Let $x$ be an eigenvector of $A$.

• Note that for all $x$, we should have $x^*Ax \geq 0$. What does this tell you about the eigenvalues of $A$?

• Note that $x^*Ax = 0 \implies x = 0$. What does this tells you about the eigenvalues of $A$?

Alternatively, if you've encountered the notion of "postive definiteness", then

Note: $A$ is (Hermitian-) positive definite if an only if the Hermitian form $f$ associated with $A$ is an inner product.