Representations and geometrical properties of generalized inverses over fields

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.

Keywords:

• Outer inverses
• inner inverses
• Moore–Penrose inverse
• $(B,C)$ inverse
• affine subspaces
• Urquhart's formula

2020 Mathematics Subject Classification:

• 15A09

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For equations (3) and (4), we assume that $\mathbb{K}$ is endowed with an involutory automorphism φ so that, for a matrix M over $\mathbb{K}$, $M^{\ast }$ denotes the transpose of the matrix $\phi (M)$ (see [27] for further details). If $\mathbb{K}$ is a subfield of the field $\mathbb{C}$ of the complex numbers, φ is assumed to be the usual complex number conjugation.

2 In [34], the field is not explicitly stated, and in [1] the proof is for complex matrices. However, one can check that the proof is valid over any field.

Additional information

Funding

Predrag Stanimirović and Miroslav Ćirić are supported by the Ministry of Education and Science, Republic of Serbia, [grant numbers 174013, 451-03-68/2020-14/200124]. Part of this work was developed while P. Stanimirović was visiting the University of Alcalá, in the frame of the project Giner de los Rios. J.R. Sendra and J. Sendra are partially supported by the grant PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish MICINN. A. Lastra is a member of the Research Group ASYNACS (Ref. CT-CE2019/683). Alberto Lastra is partially supported by the project PID2019-105621GB-I00 of Ministerio de Ciencia e Innovación. A. Lastra and J. R. Sendra also partially supported by Comunidad de Madrid and Universidad de Alcalá under grant CM/JIN/2019-010.