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Generalized principal pivot transform and its inheritance properties. (English) Zbl 1497.15008

The paper focuses on some properties of the generalized principal transform of block matrices.
Let \[M = \left( \begin{array}{cc} A & B \\ C & D \end{array} \right)\] be a complex square partitioned matrix. The generalized principal pivot transform of \(M\) with respect to \(A\) and with respect to \(D\) are defined as \[ \operatorname{gppt}(M,A) = \left( \begin{array}{cc} A^+ & -A^+B \\ CA^+ & D-CA^+B \end{array} \right), \] and \[ \operatorname{gppt}(M,D) = \left( \begin{array}{cc} A-BD^+C & BD^+ \\ -D^+C & D^+ \end{array} \right), \] respectively. Here, the superscript \(+\) denotes the Moore-Penrose inverse.
The paper studies some specific equalities concerning \(\operatorname{gppt}(M,A)\) and \(\operatorname{gppt}(M,D)\). For example, the authors characterize when \(\operatorname{gppt}(M,A)^+ = \operatorname{gppt}(M,D)\) and \(\operatorname{gppt}(\operatorname{gppt}(M,A),A^+) = M\). Also, some inheritance properties for generalized principal pivot transform are studied.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A04 Linear transformations, semilinear transformations
15B48 Positive matrices and their generalizations; cones of matrices
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References:

[1] A. Ben-Israel and T.N.E. Greville. Generalized inverses: volume 15 of CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC. Springer, New York, second edition, 2003. Theory and applications. · Zbl 1026.15004
[2] Bisht, K.; Ravindran, G.; Sivakumar, KC, Pseudo Schur complements, pseudo principal pivot transforms and their inheritance properties, Electronic Journal of Linear Algebra, 30, 455-477 (2015) · Zbl 1326.15007
[3] Choudhury, PN; Sivakumar, KC, Tucker’s theorem for almost skew-symmetric matrices and a proof of Farkas’ lemma, Linear Algebra and its Applications, 482, 55-69 (2015) · Zbl 1321.15059
[4] McDonald, JJ; Psarrakos, PJ; Tsatsomeros, MJ, Almost skew-symmetric matrices, Rocky Mountain Journal of Mathematics, 34, 1, 269-288 (2004) · Zbl 1058.15029
[5] Meenakshi, AR, Principal pivot transforms of an \(EP\) matrix, Comptes Rendus Mathématiques de l’Académie des Sciences Canada, 8, 2, 121-126 (1986) · Zbl 0598.15006
[6] Kannan, MR; Bapat, RB, Corrigendum to “Generalized principal pivot transform” [Linear Algebra Appl. 454 (2014) 49-56] [mr3208408], Linear Algebra and its Applications, 459, 620-621 (2014) · Zbl 1332.15015
[7] Kannan, MR; Bapat, RB, Generalized principal pivot transform, Linear Algebra and its Applications, 454, 49-56 (2014) · Zbl 1291.15005
[8] Kannan, MR; Sivakumar, KC, \(P_{\dagger }\)-matrices: a generalization of \(P\)-matrices, Linear Multilinear Algebra, 62, 1, 1-12 (2014) · Zbl 1318.15003
[9] Sivakumar, KC, A class of singular \(R_0\)-matrices and extensions to semidefinite linear complementarity problems, Yugoslav Journal of Operations Research, 23, 2, 163-172 (2013) · Zbl 1299.90349
[10] Tsatsomeros, MJ, Principal pivot transforms: properties and applications, Linear Algebra and its Applications, 307, 1-3, 151-165 (2000) · Zbl 0998.15006
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