TITLE: Optimization of a function of a positive definite matrix and its inverse
QUESTION [5 upvotes]: 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

REPLY [2 votes]: You are actually looking to solve the continuous algebraic Riccati equation. For convenience, I will write your $B$ as $X=X^T$. Then the equation you're trying to solve is simply
$$ X - XAX + (-C) = 0$$
Or even more explicitly, writing the Cholesky factorization of $A=BB^T$
$$ \left( \frac{1}{2}I \right)^TX + X\left( \frac{1}{2}I \right) - XBB^TX + (-C) = 0$$
The solution of an algebraic Riccati equation using Hamiltonian matrices is a standard topic in control theory. But I would instead refer you to the "care" command in MATLAB.