Construction of sign k-potent sign patterns and conditions for such sign patterns to allow k-potence

TL;DR

This work addresses constructing sign patterns with prescribed idempotent or potence behavior and determining when such patterns allow $k$-potence. It introduces a new single-iteration algorithm for all sign idempotent patterns and a structured algorithm for sign $k$-potent patterns in cyclic normal form, with termination guarantees under mild structural constraints. It provides necessary and sufficient conditions for a sign $k$-potent pattern to allow $k$-potence, including decomposition into irreducible components and the PPO framework, thereby unifying and extending a line of results by Eschenbach, Huang, Park–Pyo, Stuart, and Lee–Park. The findings yield constructive tools for classifying and synthesizing sign patterns with specified power behavior, with implications for qualitative matrix theory and combinatorial matrix analysis.

Abstract

A sign pattern is a matrix whose entries are from the set $\{+,-, 0\}$. A square sign pattern $A$ is called sign $k$-potent if $k$ is the smallest positive integer for which $A^{k+1}=A$, and for $k=1$, $A$ is called sign idempotent. In 1993, Eschenbach \cite{01} gave an algorithm to construct sign idempotent sign patterns. However, Huang \cite{02} constructed an example to show that matrices obtained by Eschenbach's algorithm were not necessarily sign idempotent. In \cite{03}, Park and Pyo modified Eschenbach's algorithm to construct all reducible sign idempotent sign patterns. In this paper, we give an example to establish that the modified algorithm by Park and Pyo does not always terminate in a single iteration; the number of iterations, depending on the order of the sign pattern, could be large. In this paper, we give a new algorithm that terminates in a single iteration to construct all possible sign idempotent sign patterns. We also provide an algorithm for constructing sign $k$-potent sign patterns. Further, we give some necessary and sufficient conditions for a sign $k$-potent sign pattern to allow $k$-potence.

Theorems & Definitions (43)

• Theorem 1.1: Theorem 1.4, 01
• Theorem 2.1: Theorem 10, 05
• Theorem 2.2: Theorem 11, 05
• Lemma 2.1: Lemma 7, 06
• Theorem 2.3: Theorem 9, 06
• Lemma 3.1: Lemma 1.1, 02
• Lemma 3.2: Lemma 1.2, 02
• Lemma 3.3: Lemma 1.2(ii), 02
• Example 3.1
• Theorem 3.1