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

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SUMMARY

The maximum trace of a complex N x N matrix A when evaluated over the unitary group U(N) is defined as maxV ∈ U(N) |Tr(AV)| = Tr(√(AA†)). Additionally, it has been established that maxV ∈ U(N) Re(Tr(AV)) = Tr(√(AA†)) as well. This conclusion simplifies the evaluation of the real part of the trace for unitary matrices.

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  • Understanding of complex matrices and their properties
  • Familiarity with unitary matrices and the group U(N)
  • Knowledge of matrix traces and their significance
  • Basic concepts of matrix adjoints and square roots of matrices
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Mathematicians, physicists, and researchers in quantum mechanics who are dealing with complex matrices and their properties, particularly in the context of unitary transformations.

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

[tex]U(N)[/tex] is the group of unitary [tex]N \times N[/tex] matrices.

(I could not preview my post properly, so I apologize for any latex-misstakes )
 
Last edited:
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You may ignore this question. It was not very difficult to show that [tex]\max_{V \in U(N)} Re(Tr(AV))=Tr(\sqrt{AA^{\dagger}})[/tex] aswell. :blushing:
 

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