Given $A\in\mathbb{R}^{n\times n}$, what is the value for this optimization problem and can this be expressed in terms of $A$? \begin{align*} &\max \text{Tr}|AXA^T+AYA^T|+\text{Tr}|AXA^T-AYA^T|\\ \text{s.t}&~ X=X^T,Y=Y^T\\ &X^2=Y^2=I \end{align*} Here for a Hermitian or real symmetric matrix $A$, $Tr|A|=\sum_i|\lambda_i| $, $\lambda_i$ are eigenvalues of $A$
If $A=I$, then $\text{Tr}|X+Y|+\text{Tr}|X-Y|\leq\sqrt{n}(\sqrt{\text{Tr}(X+Y)^2}+\sqrt{\text{Tr}(X-Y)^2})=\sqrt{n}(\sqrt{\text{Tr}(2I+2XY)}+\sqrt{\text{Tr}(2I-2XY)})\leq 2\sqrt{2}n$, the equality holds if and only if $\text{Tr}(XY)=0$
I tried the method in $\max{\text{Tr}|AXA^{T}+BXB^{T}|}$ where $X^2=I$. Suppose $|AXA^T+AYA^T|=(AXA^T+AYA^T)U_1,|AXA^T-AYA^T|=(AXA^T-AYA^T)U_2$, \begin{align*} &\text{Tr}|AXA^T+AYA^T|+\text{Tr}|AXA^T-AYA^T|\\ =&\text{Tr}(\begin{bmatrix}AX&AY\end{bmatrix} \begin{bmatrix}A^T(U_1+U_2)\\ A^T(U_1-U_2)\end{bmatrix}) \end{align*} The problem here is $U_1+U_2$ and $U_1-U_2$ may not be unitary, thus not easy to go through.
