Is there a name for the matrix equation A X B + B X A + C X C = D? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:05:02Z http://mathoverflow.net/feeds/question/6166 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6166/is-there-a-name-for-the-matrix-equation-a-x-b-b-x-a-c-x-c-d Is there a name for the matrix equation A X B + B X A + C X C = D? Jiahao Chen 2009-11-19T20:29:41Z 2013-05-14T22:02:21Z <p>I happen to be working on a problem that reduces to solving the following equation:</p> <p>$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$</p> <p>where <strong>A</strong> through <strong>D</strong> are known matrices ( <strong>A</strong>, <strong>B</strong>, <strong>D</strong> are real, symmetric matrices and <strong>C</strong> is real and antisymmetric), and <strong>X</strong> is an unknown square matrix to be solved for.</p> <p>Is there a name for this equation, and is there any known algorithm for solving this equation? (Without the <strong>C X C</strong> term this reduces to the continuous Lyapunov equation given either <strong>A</strong> or <strong>B</strong> is an invertible matrix. I wonder if anyone working in control theory may have seen such equations before.)</p> http://mathoverflow.net/questions/6166/is-there-a-name-for-the-matrix-equation-a-x-b-b-x-a-c-x-c-d/6207#6207 Answer by Greg Kuperberg for Is there a name for the matrix equation A X B + B X A + C X C = D? Greg Kuperberg 2009-11-20T02:01:41Z 2009-11-20T02:01:41Z <p>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 <a href="http://mathoverflow.net/questions/5209/notions-of-matrix-differentiation" rel="nofollow">matrix differentiation</a>, 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).</p> http://mathoverflow.net/questions/6166/is-there-a-name-for-the-matrix-equation-a-x-b-b-x-a-c-x-c-d/6211#6211 Answer by Ho Chung Siu for Is there a name for the matrix equation A X B + B X A + C X C = D? Ho Chung Siu 2009-11-20T03:09:35Z 2009-11-20T03:09:35Z <p>Another suggestion is to reduce the equation to a linear one, but I'm not sure if it's a practical method.</p> <p>One may use the <a href="http://en.wikipedia.org/wiki/Tensor%5Fproduct#Kronecker%5Fproduct%5Fof%5Ftwo%5Fmatrices" rel="nofollow" title="Kronecker product of two matrices">Kronecker product of two matrices</a> 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.</p> <p>More reference: V.V. Prasolov, Problems and Theorems in Linear algebra 27.5 (p.123)</p> http://mathoverflow.net/questions/6166/is-there-a-name-for-the-matrix-equation-a-x-b-b-x-a-c-x-c-d/64126#64126 Answer by Pait for Is there a name for the matrix equation A X B + B X A + C X C = D? Pait 2011-05-06T13:50:42Z 2011-05-06T13:50:42Z <p>This is a linear equation. As such, it is not hard to solve numerically for specific values of $A, B, C,$ and $D$. </p> <p>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.</p> http://mathoverflow.net/questions/6166/is-there-a-name-for-the-matrix-equation-a-x-b-b-x-a-c-x-c-d/64139#64139 Answer by Federico Poloni for Is there a name for the matrix equation A X B + B X A + C X C = D? Federico Poloni 2011-05-06T17:27:20Z 2011-05-06T17:27:20Z <p>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"). </p> <p>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.</p>