I want prove the formula below be to an inner product:
Given two non-zero vectors $u$ and $v$, with $A$ symmetric and positive definite,
$$\langle u,v\rangle_A:=\langle Au,v\rangle=\langle u,A^Tv\rangle=\langle u,Av\rangle=u^TAv$$
I know we must check $\langle u,v\rangle_A$ satisfies 3 properties:
Linear in the first argument.
But I don't know what to do with matrix. Please help me prove this definition is an inner product.