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.

 

[blue_raver22]

I can assure you that the general form of an inner product on \mathbf{C}^n is indeed as stated in your question. This form is widely accepted and used in mathematics and physics, and it has been rigorously proven to satisfy all the properties of an inner product. The proof for this can be found in various mathematical texts and journals, such as "Linear Algebra" by David C. Lay and "Inner Product Spaces" by Hilbert and Schmidt. Additionally, there are numerous online resources and lectures available that discuss and demonstrate this general form of an inner product. I recommend consulting with your mathematics or physics professor for further clarification and guidance on the topic.