Representations and geometrical properties of generalized inverses over fields. Issue 22 (21st December 2022)

• Record Type: Journal Article
• Authors: Stanimirović, Predrag S.; Ćirić, Miroslav; Lastra, Alberto; Sendra, J. Rafael; Sendra, Juana
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 22(2022)
• Page Start: 7318
• Page End: 7338
• Publication Date: 2022-12-21
• Subjects: Outer inverses -- inner inverses -- Moore–Penrose inverse -- (B, C) inverse -- affine subspaces -- Urquhart's formula; 15A09; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2021.1985420
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : In this paper, as a generalization of Urquhart's formulas, we present a full description of the sets of inner inverses and ( B, C ) -inverses over an arbitrary field. In addition, identifying the matrix-vector space with an affine space, we analyse geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of ( B, C ) -inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace, but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.