This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.

Suppose I have two real, positive definite (square) matrices $\mathbf{A}$ and $\mathbf{C}$, and I wish to find another real, positive definite matrix $\mathbf{B}$ such that $\mathbf{A B} + \mathbf{B}^{-1}\mathbf{C}$ is as close as possible to identity.

I'll entertain any reasonable definition of "close" that makes the problem tractable. Maybe minimization of $\| \mathbf{A B} + \mathbf{B}^{-1}\mathbf{C} - \mathbb{I} \|$ for some choice of norm.

Does anyone have any insight or experience with such a problem?

Thanks,

Nick