If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$

Question

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$

I am studying now bilinear form .I wanted to prove above theorem.

I know that for bilinear for to represent dot product It's matrix is of form $P^TP$ which provide reverse direction.

I not able to prove forword direction. Any hint will be appreciated

Answer 1

Hint: You must show

• $\langle X,X\rangle\geq 0$ and $\langle X,X\rangle= 0 \iff X=0$
• $\langle X,Y\rangle=\langle Y,X\rangle$
• $\langle X+Y,Z \rangle=\langle X,Z\rangle+\langle Y,X\rangle$ and $\langle \alpha X,Y\rangle=\alpha\langle X,Y\rangle,\quad\alpha\in\mathbb{R}^n$.