The weak core inverse
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Aequationes Mathematicae
The weak core inverse D. E. Ferreyra, F. E. Levis, A. N. Priori, and N. Thome
Abstract. In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681–697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu. Mathematics Subject Classification. 15A09. Keywords. Generalized inverses, Core inverse, Weak group inverse, Core EP decomposition.
1. Introduction The classical Moore–Penrose inverse [22] and Drazin inverse [10] were defined in the fifties and have been thoroughly studied since then. On the other hand, generalized inverses such as core inverses [2], BT inverses [3], core EP inverses [18], DMP inverses [16], CMP inverses [19], WG inverses [27], etc., were introduced in the last decade and, nowadays, they attract the attention of many researchers. In contrast to the classical ones, these recent generalized inverses (from 2010 onwards) allow us to tackle new problems and are opening up new horizons in this field both theoretical and applied. Generalized inverses of matrices are applied in areas as varied as Markov chains [4], coding theory [29], chemical equations [24], robotics [9], geology [5], etc. Matrix partial orders is another important area in which generalized inverses are an essential tool towards which attention is directed [7,8,20,25,32]. Because the core inverse was defined only for the class of index-one matrices and since the aforementioned extensions have enhanced their understanding, there is an obvious desire to extend it to new arbitrary-index classes. Motivated
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D. E. Ferreyra et al.
by these facts, our main aim is to introduce and investigate a new generalized inverse, namely the weak core inverse. Let us now recall notions of several generalized inverses and notations. We denote the set of all m × n complex matrices by Cm×n . For A ∈ Cm×n , the symbols A∗ , A−1 , rk(A), N (A), and R(A) will stand for the conjugate transpose, the inverse (m = n), the rank, the kernel, and the range space of A, respectively. Moreover, In will refer to the n × n identity matrix. A matrix X ∈ Cn×m that satisfies the equality AXA = A is called an inner inverse or {1}-inverse of A, and a matrix X ∈ Cn×m that satisfies the equality XAX = X is called an outer inverse or {2}-inverse of A. An n × m matrix X satisfying AXA = A and XAX = X is called a reflexive inverse or {1, 2}-inverse of A. For A ∈ Cm×n , the Moore–Penrose inverse of A is the unique matrix † A ∈ Cn×m satisfying the following four equations [1] AA† A = A,
A† AA† = A† ,
(AA† )∗ = AA† ,
(A† A)∗ = A† A.
The Moore–Penrose inverse can be used to represent o
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