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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:

  1. Conjugate symmetric.

  2. Linear in the first argument.

  3. Positive definite.

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

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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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