Group Theory, unitary representation and positive eigenvalues

[Aradan]

Hi, I'm new in this forum.
I have a problem i can't solve and searching on Google i couldn't find anything.
It says:


If D(g) is a representation of a finite group of order $n$, show that
$K = \sum^{i=1}_{n} D^{\dagger} (g_i) D(g_i)$ has the properties:

b) All eigenvalues of $K$ are positive


This is to prove that every representation is equivalent to a unitary representation (the problem is from the book of Matthews and Walker, problem 16-22).
I know that $K = K^{\dagger}$ implies that all eigenvalues of $K$ are real, but i can't figure it out how to demonstrate that they are positive also.


Thanks in advance (sorry for muy bad english)

 

[Citan Uzuki]

Show that the function $$\langle x,y \rangle = y^{\dagger}Kx$$ is an inner product. This will imply that K is positive-definite, and hence has only positive eigenvalues.

 

[Aradan]

Thanks for your help :)

 

[LagrangeEuler]

Sorry. I did not understand the answer.
If $K=\sum^n_{i=1}D^{\dagger}(g_i)D(g_i)$
How you know that $K$ is positive definite?