# Solution to a Matrix equation

Is there a general solution to the following matrix equation.

$A - BAB^T = C$
where B is known but can be any non-symmetric square matrix, C is known and invertible, all are n by n matrices. Is there a solution to A? or we need to use numerical methods?

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At worst it is just system of linear equations with the $n^2$ entries of $A$ as unknowns. –  Henning Makholm Sep 5 '12 at 17:28

This is a discrete Lyapunov equation, and stable, accurate numerical methods exist to solve it (for certain conditions on $C$).
Thanks for pointing in the right direction. Just found after some search that an analytical solution is possible by expressing the equation in a slightly different form. Stacking the columns of A and C we get vectors A1 and C1 and the system can be now written as $(I-B\otimes B)A1 =C1$ ($\otimes$-denote Kronecker product). This sounds right to me. However, I will have to verify. –  overloaded Sep 5 '12 at 18:26