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?

 

[blue_raver22]

Hello and welcome to the forum! Group theory is a branch of mathematics that deals with the study of symmetry in objects and systems. It has a wide range of applications in various fields, including physics, chemistry, and computer science. A representation of a group is a way of associating matrices or linear transformations with elements of the group. Unitary representation refers to a type of representation where the matrices or transformations are unitary, meaning they preserve the norm of vectors.

In this problem, you are asked to show that the sum of the product of the Hermitian adjoint of a representation and the representation itself, denoted by K, has all positive eigenvalues. This is important because it implies that every representation of a finite group can be transformed into a unitary representation. This is a fundamental result in group theory and has important implications in many areas of mathematics and physics.

To prove that all eigenvalues of K are positive, we can use the fact that K is a Hermitian matrix, meaning it is equal to its own Hermitian adjoint. This implies that all eigenvalues of K are real. To show that they are positive, we can use the concept of positive definite matrices. A matrix is positive definite if all its eigenvalues are positive. Since K is Hermitian, it can be diagonalized by a unitary transformation, meaning it can be written as K = UΛU^†, where U is a unitary matrix and Λ is a diagonal matrix with the eigenvalues of K on its diagonal.

Now, since K is Hermitian, its eigenvalues are real, and since it is also positive definite, its eigenvalues are positive. This means that all the eigenvalues of K are positive, as desired. Therefore, we have shown that the sum of the product of the Hermitian adjoint of a representation and the representation itself has all positive eigenvalues, which implies that every representation of a finite group is equivalent to a unitary representation.

I hope this explanation helps you understand the problem better. If you have any further questions, please feel free to ask. Good luck with your studies!