A Generalization of the SMW Formula of OperatorA+YGZ*to the{2}-Inverse Case

Abstract

The classical Sherman-Morrison-Woodbury (for short SMW) formulaA+YGZ*-1=A-1-A-1YG-1+Z*A-1Y-1Z*A-1is generalized to the{2}-inverse case. Some sufficient conditions under which the SMW formula can be represented asA+YGZ*-=A--A-YG-+Z*A-Y-Z*A-are obtained.

Extension of the GSMW Formula in Weaker Assumptions

“…According to Theorem 3 in this paper, Theorem 5 and Corollary 6 in [9] still hold under weaker assumptions. It must be noted that there are no assumptions on ⊙ in Theorem 3; hence, it also present more convenience than Theorem 3 and Corollary 4 in [9] in applications. The results are even robust for the finite dimensional case.…”

“…Because of its wide applications in statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations (see [5]), the properties and generalizations of the SMW formula have caught mathematicians attention (see [1][2][3][4][5][6][7][8]). Duan (see [9]) finally generalized the SMW formula to the {2}-inverse (hence, to all the inverses, uniformly denoted by ⊙ ). Under some sufficient conditions (see [9]), the generalized Sherman-Morrison-Woodbury (for short GSMW) formula has the form…”

“…Duan (see [9]) finally generalized the SMW formula to the {2}-inverse (hence, to all the inverses, uniformly denoted by ⊙ ). Under some sufficient conditions (see [9]), the generalized Sherman-Morrison-Woodbury (for short GSMW) formula has the form…”