Does anyone know of any tricks to evaluate the following inverse of matrix sums?
$$ \left( \sum_{i=1}^m D_i A D_i \right)^{-1} $$
where the $D_i$ are non-singular diagonal matrices, and $A$ is a symmetric positive definite matrix common to all terms in the sum. All these matrices are square and $n \times n$. I would like to find a closed form solution to this matrix inverse.
Rephrasing a bit,
$${\bf B} := \sum_{i=1}^d \mbox{diag} ({\bf v}_i) \, {\bf A} \,\mbox{diag} ({\bf v}_i) = \sum_{i=1}^d {\bf v}_i {\bf v}_i^\top \circ {\bf A} = \left( \sum_{i=1}^d {\bf v}_i {\bf v}_i^\top \right) \circ {\bf A} = \left( \, {\bf V} {\bf V}^\top \right) \circ {\bf A} $$
where $\circ$ denotes the Hadamard product and ${\bf V}$ is the $n \times d$ matrix whose $i$-th column is ${\bf v}_i$. The inverse of ${\bf B}$ can be computed using standard methods.