Abstract
In this paper, as a generalization of Urquhart's formulas, we present a full description of the sets of inner inverses and -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 -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.
2020 Mathematics Subject Classification:
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 For equations (3) and (4), we assume that is endowed with an involutory automorphism φ so that, for a matrix M over , denotes the transpose of the matrix (see [Citation27] for further details). If is a subfield of the field of the complex numbers, φ is assumed to be the usual complex number conjugation.
2 In [Citation34], the field is not explicitly stated, and in [Citation1] the proof is for complex matrices. However, one can check that the proof is valid over any field.