[Math] Prove a definition function is inner product

inner-products

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.

Best Answer

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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