Controllability, observability and the solution of AX - XB = C

Abstract

A closed-form finite series representation of the unique solution X of the matrix equation AX − XB = C is developed. Using this representation, the image, kernel, and rank of X are related to the controllable and unobservable subspaces of the ( A , C ) and ( C , B ) pairs respectively. Bounds on the rank of X are obtainedin terms of the dimensions of these subspaces. In the case that C has unitary rank, an exact calculation of rank X is made. The generic rank of X with A , B fixed and C generic is evaluated.