Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem \begin{align} \max_{S}~&\det(A+S) \\s.t.~&\sum_{i}^{N}\sigma_i(S)\,=\,c \\&S\geq0 \end{align}where $\sigma_i(S)$ are the singular values of $S$ and $c$ is a given positive constant.

Note that I already know that this can be converted into a standard convex problem by replacing the constraint on sum of singular values by trace and also considering log-determinant. However, I am interested in a different approach.

My hunch is that following is the solution.

Let $A=U_a\Sigma_aU_a^H$ be its Eigen-decomposition (EVD). Since both $A$ and $S$ are symmetric positive semi-definite, their EVD and SVD are same. Let $S= U_s\Sigma_sU_s^H$ be its EVD which we are seeking. Then first part of the solution is $$U_s = U_a$$ It only remains find $\Sigma_s$ which contains the singular values of $S$. Substituting this in the original optimization problem, it breaks down into the simple form of finding the variables $\sigma_i(S)$ from the optimization problem \begin{align} \max_{\sigma_1(S),\dots,\sigma_N(S)}~&\prod_{i=1}^{N}(\sigma_i(A)+\sigma_i(S)) \\s.t.&\sum_{i=1}^N\sigma_i(S)=c \end{align} This is also equivalent to maximizing the $\log$ of the objective since the objective is a increasing function of each variable. Let us substitute $$x_i=\frac{\sigma_i(S)}{\sigma_i(A)}$$ Also, we can use the fact that determinant is the product of singular values for a symmetric positive definite matrix. Combining all this, we have the optimization problem \begin{align}\max_{x_i}\sum_{i=1}^{N}~&\log(1+x_i)+\log\det(A)\\s.t.~&\sum_{i=1}^{N}\sigma_i(A)x_i\,=\,c\end{align} The constant term in the objective can be neglected. This is an instance of a very famous problem in wireless communications where one tries to maximize the capacity of set of $N$ wireless channels subject to power constraints in each. Its solution is very well known and is typically referred to as the water filling solution. So I know the process after reaching this point. Note that all this was facilitated by the assumption that $U_s = U_a$. Is that true? How do I prove this?. If not, what are other approaches I can try?