I happen to be working on a problem that reduces to solving the following equation:
$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$
where A through D are known matrices ( A, B, D are real, symmetric matrices and C is real and antisymmetric), and X is an unknown square matrix to be solved for.
Is there a name for this equation, and is there any known algorithm for solving this equation? (Without the C X C term this reduces to the continuous Lyapunov equation given either A or B is an invertible matrix. I wonder if anyone working in control theory may have seen such equations before.)
Best Answer
Another suggestion is to reduce the equation to a linear one, but I'm not sure if it's a practical method.
One may use the Kronecker product of two matrices to rewrite the equation as $(B^T \otimes A + A^T \otimes B + C^T \otimes C)X = D$, which is a linear equation. So if the matrices aren't large, I guess one can just compute the Kronecker product directly and use Gaussian elimination to solve it.
More reference: V.V. Prasolov, Problems and Theorems in Linear algebra 27.5 (p.123)