8

It is called a T-Sylvester equation, or *-Sylvester equation in the complex case. Solvability conditions and a pseudocode algorithm based on the Schur form are in https://doi.org/10.13001/1081-3810.1479 . Analogous considerations for a more general class of equations and a Fortran-90 implementation of the last step of the resulting solution algorithm (the ...


2

The inverse of the (1,1) block of $$ \begin{bmatrix} A & B\\ C & D \end{bmatrix}^{-1} $$ is $A-BD^{-1}C$ (Schur complement). This is what you are trying to compute, if I understand correctly from your explanation ("marginalize" may be standard in your domain, but it is not standard linear algebra language). So at least you can reduce to ...


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