Orthogonal realizations of random sign patterns and other applications of the SIPP

Pub Date : 2022-12-10 DOI:10.13001/ela.2023.7579
Zachary Brennan, Christopher Cox, Bryan A. Curtis, Enrique Gomez-Leos, Kimberly P. Hadaway, L. Hogben, Conor Thompson
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引用次数: 0

Abstract

A sign pattern is an array with entries in $\{+,-,0\}$. A real matrix $Q$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A. Curtis and B.L. Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228-259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, $5\times n$ nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP.
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随机符号模式的正交实现及SIPP的其他应用
符号模式是一个包含$\{+,-,0\}$的数组。一个实矩阵$Q$是行正交的,如果$QQ^T = I$。强内积性质(SIPP),在[B.A.Curtis和B.L. Shader,正交矩阵的符号模式和强内积性质,线性代数应用,592:228-259,2020],是确定符号模式是否允许行正交的重要工具,因为它保证了附近有一个具有相同性质的矩阵,允许零项被扰动到非零项,同时保留每个非零项的符号。本文利用SIPP开始研究随机符号模式允许高概率行正交的条件。在先前工作的基础上,确定了最小限度允许正交性的$5\乘以n$ nowhere零符号模式。建立了符号模式中零项的条件,保证具有这种符号模式的任何行正交矩阵具有SIPP。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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