General form of an inner product on C^n proof

[andresordonez]

Hi, I read that the general form of an inner product on $$\mathbf{C}^n$$ is:

$$\langle \vec{x} , \vec{y} \rangle = \vec{y}^* \mathbf{M} \vec{x}$$

I see that it has what it takes to be an inner product, but it seems quite hard to demonstrate that this is the general form. Is there such a demostration? where?

Thanks!

 

[Fredrik]

You need to use the fact that the inner product is sesquilinear (linear in one of the variables and antilinear in the other), and the relationship between linear operators and matrices described here.

 

[arkajad]

Let $$e_i$$ be the standard basis of $$\mathbf{C}^n$$.
Define $$M_{ji}=\langle e_i,e_j\rangle$$ and check that your formula holds.

 

[andresordonez]

Thanks.