Is there a name for the matrix equation A X B + B X A + C X C = D? - MathOverflow

Is there a name for the matrix equation A X B + B X A + C X C = D?

2009-11-19T20:29:41Z

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.)

Answer by Greg Kuperberg for Is there a name for the matrix equation A X B + B X A + C X C = D?

2009-11-20T02:01:41Z

I'm not sure about names for this equation. As for solving it, I can say this much: It is a linear system and there is a solution in which $X$ is also symmetric. Following basics of matrix differentiation, it is the critical point of the functional $$\mathrm{Tr}(AXBX) + \frac{\mathrm{Tr}(CXCX)}2 = \mathrm{Tr}(DX).$$ This is not generally positive definite. If $A$ and $B$ are positive definition and the $C$ term is absent or small, then it is positive definite and you can use convex minimization methods (such as conjugate gradient) to solve for $X$. But in the general case, no such luck, although it simplifies matters somewhat that it is a symmetric linear system (with respect to the inner product $\langle X,Y \rangle = \mathrm{Tr}(XY)$ for symmetric matrices).

Answer by Ho Chung Siu for Is there a name for the matrix equation A X B + B X A + C X C = D?

2009-11-20T03:09:35Z

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)

Answer by Pait for Is there a name for the matrix equation A X B + B X A + C X C = D?

2011-05-06T13:50:42Z

This is a linear equation. As such, it is not hard to solve numerically for specific values of $A, B, C,$ and $D$.

As for a "closed form" solution, using matrix exponentials and the like, as in the Lyapunov equations of control theory.... I don't think there will be one except in particular cases (if $A$ and $C$ commute for instance). The eigenvectors of the operator $AXB+BXA$ can be written in terms of the eigenvectors of $A$ and $B$. I don't think it is so when $C$ is present, in general.

Answer by Federico Poloni for Is there a name for the matrix equation A X B + B X A + C X C = D?

2011-05-06T17:27:20Z

Apart from very special cases (something commuting with something else), as far as I know there is no efficient algorithm for this kind of equations with more than two summands. (by "efficient" I mean "better than the Kronecker product approach").

May sound strange, but I would actually suggest solving the Kronecker product system with an iterative method like SYMMLQ, or CG if it's positive definite. Matrix-vector products cost "only" $O(n^3)$, and dropping a term provides a better-than-nothing preconditioner.