How can I solve this matrix equation?

Question

I have a Matrix equation of the form $A-BBAB^{T}B^{T}=BCB^{T}+BBCB^{T}B^{T}$ and I'm trying to solve for $A$. It's also the case that $B$ is stochastic if that helps at all. If I multiply by inverses of $B$ to free the second term then I trap the first term and I'm not really sure what else to try. Is there a way to solve this type of equation analytically?

Answer 1

You assume that $B$ is invertible. Then the equation is in the form $A(B^{-T})^2-B^2A=U$. It is a (linear) Sylvester equation. Let $spectrum(B)=((\lambda_i)_i)$; the solution in $A$ is unique when for every $i,j$ $1/\lambda_i^2-\lambda_j^2\not= 0$, that is, when $\lambda_i\lambda_j\not=\pm 1$.

EDIT. Let $f:A\rightarrow A(B^{-T})^2-B^2A$. Since $B$ is stochastic, $1$ is an eigenvalue of $B$ and $dim(\ker(f))\geq 1$; generically $dim(\ker(f))= 1$; in particular assume that $B$ is also positive: then $1$ is an eigenvalue of $B$ of multiplicity $1$ and the other $\lambda_i$ have modulus $<1$, and finally, $dim(\ker(f))= 1$.

Conclusion. Without any hypothesis about $C$, the considered equation has generically no solutions in the unknown $A$.