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On semimonotone matrices of exact order two

Bharat Pratap Chauhan, Dipti Dubey

Abstract

In this paper, we introduce the notion of (strictly) semimonotone matrices of exact order $k$, where $0\leq k\leq n$, and explore their properties. We fully characterize the $3 \times 3$ (strictly) semimonotone matrices of exact order $2$, and show that the class of $3 \times 3$ semimonotone matrices of exact order $2$ forms a subclass of inverse $\mathbf{Z}$-matrices. We further investigate $ n\times n$ (strictly) semimonotone matrices of exact order $2$, with emphasis on their identification and construction, and establish that every $n\times n$ semimonotone $\mathbf{Z}$-matrix of exact order $2$ is invertible. Additionally, we show that when $n-k=1$, the class of (strictly) semimonotone matrices of exact order $k$ is a subclass of $\mathbf{Z}$-matrices.

On semimonotone matrices of exact order two

Abstract

In this paper, we introduce the notion of (strictly) semimonotone matrices of exact order , where , and explore their properties. We fully characterize the (strictly) semimonotone matrices of exact order , and show that the class of semimonotone matrices of exact order forms a subclass of inverse -matrices. We further investigate (strictly) semimonotone matrices of exact order , with emphasis on their identification and construction, and establish that every semimonotone -matrix of exact order is invertible. Additionally, we show that when , the class of (strictly) semimonotone matrices of exact order is a subclass of -matrices.
Paper Structure (6 sections, 13 theorems, 34 equations)

This paper contains 6 sections, 13 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semimonotone Matrices of Exact Order 2
  4. Semimonotone Matrices of Exact Order Two in $\mathbb{R}^{3 \times 3}$
  5. Semimonotone Matrices of Exact Order Two in $\mathbb{R}^{n \times n}$
  6. Semimonotone Matrices of Exact Order k

Key Result

Theorem 2.1

A matrix $A \in \mathbb{R}^{2 \times 2}$ is almost (strictly) semimonotone if and only if $A = $ where $b, c > 0$, $a, d \geq 0$ (resp., $a, d > 0$), and $\det A < 0$ (resp., $\det A \leq 0$).

Theorems & Definitions (33)

  • Theorem 2.1: wendler2019almost, Theorem 3.3
  • Theorem 2.2: tsatsomeros2019semimonotone, Corollary 5.4
  • Theorem 2.3: chauhan2023almost, Theorems 3.6 and 3.7
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • ...and 23 more