Sylvester equation
= Sylvester equation
{c}
{title2=$AX-XB=C$}
{wiki}
For complex square <matrices> $A\in M_m(\mathbb C)$ and $B\in M_n(\mathbb C)$, the Sylvester equation
$$
AX-XB=C
$$
has 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.