engleski

Sherman-Morrison-Woodbury formula for Sylvester and $T$-Sylvester equation with applications

In this paper we present the Sherman-Morrison-Woodbury-type formula for the solution of the Sylvester equation of the form \[(A_0+U_1V_1)X+X(B_0+U_2V_2)=E, \] as well as for the solution of the $T$-Sylvester equation of the form \[ (A_0+U_1V_1)X+X^T(B_0+U_2V_2)=E, \] 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 Sylvester and $T$-Sylvester equations by using these formulas and illustrate their application in several examples.

Sylvester equation; $T$-Sylvester equation; Sherman-Morrison-Woodbury formula

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