Sherman–Morrison–Woodbury formula for Sylvester and T-Sylvester equations with applications

Abstract

In this paper, we present the Sherman–Morrison–Woodbury-type formula for the solution of the Sylvester equation of the form as well as for the solution of the T-Sylvester equation of the form where U 1, U 2, V 1, V 2 are low-rank matrices. Although the matrix version of this formula for the Sylvester equation has been used in several different applications (but not for the case of a T-Sylvester equation), we present a novel approach using a proper operator representation. This novel approach allows us to derive a matrix version of the Sherman–Morrison–Woodbury-type formula for the Sylvester equation as well as for the T-Sylvester equation which seems to be new. We also present algorithms for the efficient calculation of the solution of structured Sylvester and T-Sylvester equations by using these formulas and illustrate their application in several examples.