solving matrix equation I want to solve the following matrix equation. Could anyone give me a hand? Thanks.
Given an $n \times n$ matrix $\mathbf A$ (diagonally dominant), I need to solve an $n \times n$ symmetric matrix $\mathbf  X$ such that
$\mathbf A\mathbf X+\mathbf X\mathbf A^\top =\mathbf  I$,
where $\mathbf I$ is an $n \times n$ identity matrix.

What you have is a special case of a Sylvester equation. The wiki page I linked to describes both the eigenvalue and Kronecker product approaches to solving the equation, but in practice, one uses algorithms like the one by Bartels and Stewart.
