Unitary matrix in trace and log function

Question

I am trying to do a unitary transformation $U$ on square matrix $A$ which is embedded inside a trace and natural log function, and the following property is supposed to hold:

$\mathrm{tr} (\ln (A)) = \mathrm{tr} (\ln (UAU^\dagger))$

What property of the $\mathrm{tr}$ and $\ln$ functions would allow us to do that?

Answer 1

Trace is invariant under cyclic permutations. That is, $tr(ABC) = tr(CAB) = tr(BCA)$. Trace is also linear.

Now, plug in the series for $\log$ for the right hand side, use the fact that $(U A U^+)^n = U A^n U^+$ and linearity of trace to get a sum of $tr(U A^n U^+) = tr (A^n U^+ U) = tr(A^n)$. Then, match that to the expansion you get for tr(ln(A)) in the same way.

Answer 2

A corollary of Jacobi's formula for invertible matrices is $${\rm tr}\log X = \log\det X$$ Applying it to this problem yields $$\eqalign{ {\rm tr}\log(UAU^{\dagger})&= \log\det(UAU^{\dagger}) \cr &= \log\det(A) \cr &= {\rm tr}\log(A) \cr }$$