Special solution to the Sylvester equation

Question

I'm focusing on this particular kind of Sylvester's equation: \begin{equation}AX=XA^\dagger\end{equation} where I would like that the solution $X$ defines an inner product, namely it is Hermitian and positive semidefinite. The size of $A$ is around 100, so the Kronecker product trick runs very slow. I would like to know if there is a special algorithm to this problem?

Answer 1

“I would like that the solution $X$ defines an inner product, namely it is Hermitian and positive semidefinite.”

It isn’t clear whether you want $X$ to be positive definite or positive semidefinite. If $X$ is merely positive semidefinite, it only defines a semi-inner product, not necessarily an inner product.

If you want $X$ to be positive definite, note that $AX=XA^\dagger$ implies that $A=HX^{-1}$ for some Hermitian matrix $H$. Hence $A$ must be similar to the Hermitian matrix $X^{-1/2}HX^{-1/2}$. In other words, the equation is solvable only if $A$ is a diagonalisable matrix with a real spectrum. If this is the case, let $A=P\Lambda P^{-1}$ be a diagonalisation. Then $X=PP^\dagger$ is a solution.

If you only want $X$ to be positive semidefinite, you may take $X=0$.