TITLE: Prove a definition function is inner product
QUESTION [2 upvotes]: 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:

Conjugate symmetric.
Linear in the first argument.
Positive definite.

But I don't know what to do with matrix. Please help me prove this definition is an inner product.

REPLY [1 votes]: You prove them by using the properties of matrix multiplication. For instance, for part 2, linearity in the first argument, you want to show that
$$\langle \alpha x + y, z\rangle = \alpha \langle x, z\rangle + \langle y, z\rangle.$$
The left side is
$$(\alpha x + y)^TAz.$$
Since matrix multiplication is linear, you have
$$(\alpha x + y)^TAz=(\alpha x)^TAz + y^TAz = \alpha x^TAz + y^TAz.$$
Finally these two terms correspond to the two inner products
$$\alpha \langle x, z\rangle + \langle y, z\rangle.$$
Does that make sense?
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