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.

Keywords:

• Sylvester equation
• T-Sylvester equation
• Sherman–Morrison–Woodbury formula

2010 AMS Subject Classifications:

• 65F99
• 65Z05

Acknowledgements

The authors thank the anonymous reviewers for their valuable comments and suggestions which were helpful in improving the paper.

Notes

To the authors’ knowledge, no one has formulated the Sherman–Morrison–Woodbury formula for the Sylvester equation so far. The authors thank Volker Mehrmann for having pointed out this interesting paradox to them during the 17th ILAS Conference in Braunschweig.

The authors thank Federico Poloni for pointing out this interesting problem to them during the Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, Trogir, 2011.