Extended Irreducible Nekrasov Matrices as Subclasses of Irreducible H -Matrices
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Extended Irreducible Nekrasov Matrices as Subclasses of Irreducible H-Matrices Jianzhou Liu1 · Lixin Zhou1,2 Received: 26 February 2020 / Revised: 24 August 2020 / Accepted: 5 September 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we introduce the irreducible α-Nekrasov matrices and irreducible α-SNekrasov matrices as the extended irreducible Nekrasov matrices and we analyze the relationships among the involved matrices and irreducible H -matrices. Keywords H -matrix · Irreducible α-Nekrasov matrices · S-Nekrasov matrices · Irreducible α-S-Nekrasov matrices Mathematics Subject Classification 15A45 · 15A48 · 65F05 · 65F10 · 65F99
1 Introduction H -matrix plays an important role in scientific computation and engineering applications (see [1–6]). A well-known characterization of H -matrices is given by the fact that a matrix is an H -matrix if and only if it can be scaled to a strictly diagonally dominant matrix by a nonsingular diagonal matrix (from the right side). Thus, judging H -matrices could be translated to finding such a scaling matrix, see [9–11] and the
Communicated by Fatemeh Panjeh Ali Beik. The work was supported in part by National Natural Science Foundation of China (11971413), Guangxi Municipality Project for the Basic Ability Enhancement of Young and Middle-Aged Teachers (KY2016YB532, 2019KY0795).
B
Lixin Zhou [email protected] Jianzhou Liu [email protected]
1
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, People’s Republic of China
2
School of Science, Guilin University of Aerospace Technology, Guilin 541004, Guangxi, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
references therein. In recent years, some subclasses of H -matrices were introduced, and they had been described by checkable conditions, meaning simple functions of matrix elements only. The fact showed that, it was an effective way to find out more practical criteria for judging H -matrices in some sense (see [7,12–22]). In this paper, inspired by the idea of [7,23], we introduce two subclasses of irreducible H -matrices, i.e., irreducible α-Nekrasov matrices and irreducible α-S-Nekrasov matrices, which may be considered as the extension of irreducible Nekrasov matrices. Let Cn×n (Rn×n ) denote the set of all n × n complex (real) matrices. Let N = {1, 2, · · · , n}, and let S denote any nonempty subset of N, and S = N − S be the complement set of S in N. For A = (ai j ) ∈ Cn×n , let α ∈ (0, 1], ∅ = β ⊂ N, and let |A| = (|ai j |)n×n , A(β) stands for the sub-matrix of A lying in the rows indexed by β and the columns indexed by β, and denote Pi (A) =
|ai j |, Q i (A) =
j∈N, j=i i−1
|ai j |
j=1
j∈S, j=1
|a ji |, i ∈ N,
j∈N, j=i
R1 (A) = P1 (A), Ri (A) = h 1S (A) =
n R j (A) |ai j |, 2 ≤ i ≤ n, + |a j j | j=i+1
|a1 j |, h iS (A) =
i−1
|ai j |
j=1
h Sj (A) |a j j |
+
n
|ai j |, 2 ≤ i ≤ n.
j=i+1, j∈S
Next, we would like to recall the following definitions and lemmas. Definition 1.1 [4] Let A = (
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