I have a problem similar to but simpler than this unanswered question.
Let $\mathbf{D}$ be a $p \times p$ diagonal matrix with positive, not necessarily equal diagonal elements. Let $\mathbf{1}$ be a $p \times 1$ vector of 1's. Take $$\mathbf{A} = \mathbf{D} + \mathbf{1} \mathbf{1}'.$$ Is there a closed-form expression for the symmetric square root of the inverse of $\mathbf{A}$? That is, is there an expression for the matrix $\mathbf{B} = \mathbf{A}^{-1/2}$ such that $\mathbf{B} \mathbf{A} \mathbf{B} = \mathbf{I}_p$?
Fasi, Higham, and Liu (2023; https://doi.org/10.1137/22M1471559) provide expressions for roots of low-rank updates to a multiple of an identity matrix, which can be applied here if $\mathbf{D} = \alpha \mathbf{I}_p$ (see also this blog post from Dan MacKinlay), but I need an expression for $\mathbf{D}$ with entries that are allowed to vary.
