（1）Maximize the trace of a matrix. （2）Minimize the trace of its inverse matrix. Are (1) and (2) equal?

Question

I need to solve an optimization problem which aims to $\text{maximize}$ $\text{Trace}[(U^HU)^{-1}]$. The diagonal elements of $U^HU$ are all positive. I want to know whether it is equal to solve the problem of $\text{minimize}$ $\text{Trace}[U^HU]$?

Answer 1

In general, the answer is no. Assuming $A$ has all positive eigenvalues, making Trace(A) large means making its largest eigenvalue large; the smallest eigenvalue plays essentially no role in the optimization. Making Trace($A^{-1}$) small means making its largest eigenvalue small, i.e. making the smallest eigenvalue of A large.