Weighted quaternion core-EP, DMP, MPD, and CMP inverses and their determinantal representations

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Weighted quaternion core-EP, DMP, MPD, and CMP inverses and their determinantal representations Ivan Kyrchei1 Received: 11 March 2020 / Accepted: 26 August 2020 © The Royal Academy of Sciences, Madrid 2020

Abstract In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representations by using noncommutative column and row determinants previously introduced by the author. A numerical example to illustrate the main result is given. Keywords Weighted core-EP inverse · Weighted DMP inverse · Weighted MPD inverse · Weighted CMP inverse · Generalized inverse · Moore–Penrose inverse · Weighted Drazin inverse · Quaternion matrix · Noncommutative determinant Mathematics Subject Classification 5A09 · 15A15 · 15B33

1 Introduction In the whole article, the notations R and C are reserved for fields of the real and complex numbers, respectively. Hm×n stands for the set of all m × n matrices over the quaternion skew field H = {h 0 + h 1 i + h 2 j + h 3 k | i2 = j2 = k2 = ijk = −1, h 0 , h 1 , h 2 , h 3 ∈ R}. Hrm×n determines its subset of matrices with a rank r . For given h = h 0 +h 1 i+h 2 j+h 3 k ∈ H, the conjugate of h is h = h 0 − h 1 i − h 2 j − h 3 k. For A ∈ Hm×n , the symbols A∗ and rk(A) specify the conjugate transpose and the rank of A, respectively. A matrix A ∈ Hn×n is Hermitian if A∗ = A. The index of A ∈ Hn×n , denoted Ind A = k, is the smallest nonnegative integer such that rk(Ak+1 ) = rk(Ak ). Due to [1] the definition of the weighted Drazin inverse can be generalized over H as follows.

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Ivan Kyrchei [email protected] Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, str. Naukova 3b, Lviv 79060, Ukraine 0123456789().: V,-vol



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I. Kyrchei

Definition 1 For A ∈ Hm×n and W ∈ Hn×m , the W-weighted Drazin inverse of A with respect to a nonzero matrix W, denoted by Ad,W , is the unique solution to equations, (AW)k+1 XW = (AW)k , XWAWX = X, AWX = XWA, where k = Ind(AW). The properties of the complex W-weighted Drazin inverse can be found in [2–5]. These properties can be generalized to H. Among them, if A ∈ Hm×n with respect to W ∈ Hn×m , then 2  2  Ad,W =A (WA)d = (AW)d A. (1) Let A ∈ Hn×n and W = In be the identity matrix of order n. Then X = Ad is the Drazin inverse of A. In particular, if Ind A = 1, then the matrix X is called the group inverse and it is denoted by X = A# . Some properties and applications of the group inverse can be find in e.g. [6–8]. Using the Penrose equations [9], the Moore–Penrose inverse of a quaternion matrix can be defined as well (see, e.g. [10]). Definition 2 The Moore–Penrose inverse of A ∈ Hm×n is called the unique matrix X, denoted by A† , satisfying the following four equations AXA = A, XAX = X, (AX)∗ = AX, (XA)∗ = XA. P A := AA† and Q