Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of $S$ padded by zeros. Does there exist a matrix $\Psi$ such that
$$ tr(Y[RSR]^+)=tr(\Psi S^{-1}) $$
where $\psi$ is not a function of $S$? If so, what is $\Psi$?
Possibly relevant prior answers:
- Are there any decompositions of a symmetric matrix that allow for the inversion of any submatrix?
- Block inverse of symmetric matrices
- Computing the Moore-Penrose pseudoinverse of a submatrix
I've spent some time trying to do this with block-wise matrix inversion but so far that has been a dead end. Specifically, it is true that
$$ [RSR]^+=R S^{-1}R-R S^{-1} Q [Q S^{-1} Q]^+ Q S^{-1} R $$
where $Q=I-R$ but this doesn't seem to suggest a tractable solution.
Context: This is question is related to handling missing data in a Normal-Inverse Wishart model, with $R$ masking some of the data points.
