Max Trace of U(N) Matrices: A Complex Problem

[DavidK]

Assume $$A$$ is a complex $$N \times N$$ matrix. It is well known that $$\max_{V \in U(N)} |Tr(AV)| = Tr(\sqrt{AA^{\dagger}})$$. But what is
$$\max_{V \in U(N)} Re(Tr(AV))$$?

$$U(N)$$ is the group of unitary $$N \times N$$ matrices.

(I could not preview my post properly, so I apologize for any latex-misstakes )

 

[DavidK]

You may ignore this question. It was not very difficult to show that $$\max_{V \in U(N)} Re(Tr(AV))=Tr(\sqrt{AA^{\dagger}})$$ aswell.

 

[tacman]

The problem of finding the maximum trace of U(N) matrices, specifically the real component, is indeed a complex problem. This is because the unitary group U(N) is a highly complex and non-commutative group, making it challenging to find a closed-form solution for the maximum real trace.

However, there are some known results that can help us understand the problem better. One of these results is the fact that the maximum trace of a U(N) matrix A can be expressed as Tr(√(AA†)), as mentioned in the content. This means that the maximum real trace will be the real part of this expression, which is Tr(Re(√(AA†))).

Another way to approach this problem is to consider the eigenvalues of A. Since A is a complex matrix, its eigenvalues can be complex as well. The maximum real trace of A will then be the sum of the real parts of its eigenvalues. However, finding the eigenvalues of a U(N) matrix can also be a complex problem, as it involves solving a non-linear system of equations.

In conclusion, finding the maximum real trace of U(N) matrices can be a complex and challenging problem due to the non-commutative nature of the unitary group and the complex eigenvalues of the matrices. Further research and analysis may be needed to find a closed-form solution or efficient computational methods for this problem.