# Global minimum of a function involving matrix exponentials

I have a matrix system $$$$\vec{x}(t) = e^{(iAt)} \vec{x}(0)$$$$ that comes from the solution of the differential system $$$$\frac{d}{dt}\vec{x} = iA\vec{x},$$$$ where $$$$e^{(iAt)} = \sum_{k = 0}^{\infty} \frac{(iAt)^k}{k!}$$$$ and where A is under constraint such that $$A_{n,m} = A_{i,j}$$ if $$i+j = n+m$$. Matrix $$A$$ is a very big matrix (say of size $$200\times 200$$). I want to find values of $$A_{n,m}$$ to reach the global minimum of the cost function $$$$O = (\sqrt{Tr(e^{(iAt)})^*Tr(e^{(iAt)})} - 1)^2$$$$ where the $$(Tr(e^{(iAt)})^*$$ is the complex conjugate of the trace, if the global minimum exist. Does anyone have an idea how to tackle this problem?

• Aren’t you missing a t in the exponent? Otherwise why is x a function of t? Jul 30 at 5:36
• Just so you know the terminology, matrices satisfying $A_{n,m} = A_{i,j}$ if $i+j = n+m$ are called Hankel matrices. Jul 30 at 9:33
• That function takes complex values if I am not mistaken; is there a modulus missing? Jul 30 at 9:47
• Please resolve (edit) the inconsistency between your last Comment and the "cost" function described currently in the body of your Question. It is worth clarifying there at the same time that your Hankel matrix $A$ has complex entries. A few words about the cases when $A$ is $1\times 1$ and $2\times 2$ would be welcome confirmation that your problem formulation is well-defined.
– hardmath
Jul 31 at 3:08
• The cost function you show depends on $t$, but the objective function should not. Something is missing. Jul 31 at 3:14