I am trying to solve the following optimization problem (analytically)
$$P^* := \arg\min_{P\succeq 0} \,\,\left( (1-\beta)\log|P+I|+\beta\log|P+R| \right)$$
where all matrices are in $\mathbb{R}^{n \times n}$, $I$ is the identity matrix, $R$ is a diagonal matrix with diagonal entries $r_i \geq 1$, and $\beta \geq 0$. $P$ is restricted to be a symmetric positive semidefinite matrix. It will suffice to show that there exists a non-negative diagonal matrix which achieves the minimum.
Progress
My approach so far was to write $P$ as $P=UDU^T$ where $U$ is orthogonal and $D$ is a non-negative diagonal matrix. Taking the derivative with respect to each element of $D$ and equating to $0$ yields
$D=U^T(\beta(R-I)-R)U$
If $\beta(R−I)−R$ is positive on the diagonal, $U$ must be diagonal and the problem is solved. Otherwise, for small enough $\beta$ the optimal $P$ is singular and is found on the boundary of the PSD cone. In the latter case, I'm struggling to show that $U$ is still diagonal.
This problem arises when trying to apply the Information bottleneck method to Gaussian variables.