A question regarding matrix of inner product in the ordered basis from Hoffman Kunze

Question

While studying Chapter - Inner Product Spaces from Hoffman Kunze, I have a question in section 8.1 .

Adding it's image:

How can I derive the condition that the matrix must satisfy the additional condition that $X^{*}GX>0$ , X$\neq$ 0 .

Answer 1

This follows by definition of inner product: $(\alpha|\alpha)>0$ if $\alpha \neq 0$.