Find all skew-symmetric matrices given their anti-commutator with a symmetric matrix

Question

Let $S$ be a skew-symmetric matrix and $J$ a symmetric matrix. Is it possible to find all skew-symmetric matrices $\Omega$ satisfying $$S = J\Omega + \Omega J$$ in terms of $S$ and $J$?

Answer 1

Vectorizing the equation yields $$\eqalign{ {\rm vec}(S) &= {\rm vec}(J\Omega I) + {\rm vec}(I\Omega J) \cr s &= (I\otimes J + J\otimes I)\,\omega \,\,{\dot =}\,\, M\omega \cr \omega &= M^+s + (I-M^+M)\,a \cr \Omega &= {\rm Mat}\big(M^+s + (I-M^+M)\,a\big) \cr }$$ where ${\rm Mat}()$ is the inverse of the ${\rm vec}()$ operation, $\otimes$ represents the Kronecker product, $M^+$ is the Moore-Penrose inverse of $M$, and $a$ is an arbitrary vector.

If $M$ is full rank then $(I-M^+M)=0$ and there is only one solution, otherwise there are an infinite number, i.e. a solution for each $a$ vector.