3
$\begingroup$

Consider the following linear equation

$A X + X B = C$

where $A$, $B$, $C$, are known real $n \times n$ square matrices, and where $X$ is an unknown real $n \times n$ square matrix. I want to have an analytical expression for the matrix $X$ that solves the above linear problem.

  • 2
    What you have there is a [Sylvester equation](http://en.wikipedia.org/wiki/Sylvester_equation).2011-09-09
  • 1
    Which you can solve in Matlab via `X = lyap(A,B,C) `2011-09-09
  • 2
    But should you need to write your own solver: see [this](http://dx.doi.org/10.1145/361573.361582) and [this](https://www.cs.cornell.edu/cv/ResearchPDF/Hessenberg.Schur.Method.pdf).2011-09-09
  • 0
    @ J.M.: thank you for the references. I was looking for a analytical solution, but from what I see that does exist in simple form (not at least one where X is not reshaped in vector form).2011-09-09
  • 0
    Yeah, you'll need the Kronecker product and the $\mathrm{vec}$ operator for an "analytical solution".2011-09-09

2 Answers 2