Xiaoji Liu

Characterizations of the core inverse and the core partial ordering

In this note, we revisit the core inverse and the core partial ordering introduced by Baksalary and Trenkler [Linear Multilinear Algebra. 2010;58:681–697]. We prove that the core inverse of is the unique solution of and , and establish several characterizations of the core inverse, the core partial ordering and the reverse order law for the core inverse.

The group inverse of the combinations of two idempotent matrices

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈ ℂ and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ.

Determinantal representation of weighted generalized inverses

Abstract In this article, we introduce new determinantal representations of the weighted generalized inverse by using a lemma of Robinson [D.W. Robinson, The classical adjoint, Linear Algebra Appl. 411 (2005) 254–276]. And, using generalized Cramer’s rule, we get the method for computing minors of the weighted Moore–Penrose inverse and the W -weighted Drazin inverse.

Nonsingularity and group invertibility of linear combinations of two k-potent matrices

An n × n complex matrix A is said to be k-potent if A k = A. Let T 1 and T 2 be k-potent and c 1 and c 2 be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c 1 T 1 + c 2 T 2 of two k-potent matrices T 1 and T 2.

Integral and limit representations of the outer inverse in Banach space

We consider the various representations of the outer inverse of an operator A on Banach spaces.

Further results on the reverse order law for the group inverse in rings

In this paper, we use the Drazin inverse to derive some new equivalences of the reverse order law for the group inverse in unitary rings. Moreover, if the ring has an involution, we present more equivalences when both involved elements are EP.

New results on reverse order law for {1,2,3}- and {1,2,4}-inverses of bounded operators

In this paper, using some block-operator matrix techniques, we give necessary and sufficient conditions for the reverse order law to hold for {1, 2, 3}and {1, 2, 4}-inverses of bounded operators on Hilbert spaces. Furthermore, we present some new equivalents of the reverse order law for the Moore-Penrose inverse.

The absorption laws for the generalized inverses

Abstract In this paper, we give necessary and sufficient conditions for the absorption laws in terms of { 1 } , { 1 , 2 } , { 1 , 3 } and { 1 , 4 } -inverses. Also, we consider the various types of mixed absorption law for the generalized inverses.

Additive results for the group inverse in an algebra with applications to block operators

We derive a very short expression for the group inverse of a 1 + ··· + a n when a 1, … , a n are elements in an algebra having group inverse and satisfying a i a j  = 0 for i < j. We apply this formula in order to find the group inverse of 2 × 2 block operators under some conditions.

Integral representation of the W-weighted Drazin inverse for Hilbert space operators

This paper studies the integral representation of the W-weighted Drazin inverse for bounded linear operators between Hilbert spaces. By using operator matrix blocks, some integral representations of the W-weighted Drazin inverse for Hilbert space operators are established.