 Sylvester equation

For complex square matrices $A\in M_m(\mathbb{C})$ and $B\in M_n(\mathbb{C})$, the Sylvester equationhas a unique $m$-by-$n$ matrix solution $X$ for every $C$ exactly when the spectra of $A$ and $B$ are disjoint. The linear map $X\mapsto AX-XB$ has eigenvalues ${\alpha }_i-{\beta }_j$: triangularize $A$ and $B$, or use its Kronecker product representation $I_n\otimes A-B^T\otimes I_m$ with columnwise vectorization. It is invertible exactly when none of those differences vanishes. Over another field, the same criterion uses the polynomial roots of the two characteristic polynomials in an algebraic closure.

For one-by-one matrices, the equation is $(a-b)x=c$, making both the solution $x=c/(a-b)$ and the obstruction at $a=b$ immediate.

The Sylvester equation is a type of linear matrix equation that has the general form: \[ AX + XB = C \] where: - \(A\) and \(B\) are given matrices of appropriate dimensions, - \(X\) is the unknown matrix to be solved for, and - \(C\) is a given matrix.