Weighted binary relations involving core-EP inverse
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Weighted binary relations involving core-EP inverse Yuefeng GAO1 , Jianlong CHEN2 ,
Pedro PATR´ICIO3
1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China 2 School of Mathematics, Southeast University, Nanjing 210096, China 3 CMAT-Centro de Matem´ atica, Universidade do Minho, Braga 4710-057, Portugal
c Higher Education Press 2020
Abstract We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations. This relation becomes a pre-order. Then, one-sided preorders associated to the weighted core-EP inverse are given from two perspectives. Finally, we make a comparison for these two sets of one-sided weighted pre-orders. Keywords Weighted core-EP inverse, core-EP inverse, pseudo core inverse, pre-order MSC 15A09, 06A06 1
Introduction
A binary relation is a pre-order if it is reflexive and transitive; if it is also antisymmetric, then it is a partial order. The theory of partial orders (preorders) based on various generalized inverses has been increasingly investigated, such as ∗-partial order [2], minus partial order [7,16], sharp partial order [12], Drazin pre-order [11,13], core partial order [1,17,19], and core-EP pre-order [5,14,18]. Meanwhile, weighted Drazin pre-order and one-sided weighted Drazin pre-order were studied by Hern´andez et al. [8,9]. Motivated by the above papers, in this paper, our main goal is to study new binary relations defined by the weighted core-EP inverse. Throughout this paper, Cm×n is used to denote the set of all m × n complex matrices. For each complex matrix A ∈ Cm×n , A∗ denotes the conjugate transpose of A, and R(A) denotes the range of A. The index of A ∈ Cn×n , denoted by ind(A), is the smallest non-negative integer k for which rank(Ak ) = Received July 1, 2019; accepted August 1, 2020 Corresponding author: Yuefeng GAO, E-mail: [email protected]
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rank(Ak+1 ). Recall that the core-EP inverse was proposed by Manjunatha Prasad and Mohana [10] for a square matrix of arbitrary index, as an extension of the core inverse restricted to a square matrix of index at most 1 in [1]. Then, Gao and Chen [4] characterized the core-EP inverse (also known as the pseudo core inverse) in terms of three equations. Let A ∈ Cn×n with ind(A) = k. The † core-EP inverse of A, denoted by A , is the unique solution of the system XAk+1 = Ak ,
AX 2 = X,
(AX)∗ = AX.
The core-EP inverse is an outer inverse (resp., {2}-inverse), i.e., † † † A AA = A ,
see [4]. Lemma 1 [4]
Let A ∈ Cn×n with ind(A) = k. Then † A = AD Ak (Ak )† .
Wang introduced the core-EP pre-order as follows. Definition 1 [18]
† Let A, B ∈ Cn×n . Then A B if † † A A = A B,
† † AA = BA .
An extension of the core-EP inverse from a square matrix to a rectangular matrix was made by Ferreyra et al. [3] and was named the weighted core-EP inverse. Let A ∈ Cm×n and W ∈ Cn×m (W 6= 0) with k = max{ind(AW ), ind(W A)}. † The W -weighted core-EP inverse A ,W of A is the unique solution of
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