MATH319 Slides 126 Cases 128 An integral solution of Sylvester’s equation

127 Sylvester’s criterion

Proposition

Given $A$ and $B$ , Sylvester’s equation $AY+YB=-C$ has a unique solution $Y$ for all $C$ if and only if $A$ and $-B$ have no common eigenvalues. Remark. Fix $A$ and $B$ . Then the transformation $T:M_{n}\rightarrow M_{n}:$ $Y\mapsto AY+YB$ is linear on the finite-dimensional vector space $M_{n}$ , so either:

(i) $T$ is invertible, and for all $C$ there exists a unique $Y$ such that $T(Y)=-C$ ; or

(ii) $T$ is not invertible and hence does not have full rank, so $T(Y)=-C$ has no solution for some $C$ .