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

TL;DR

This paper explores conditions under which random sign patterns permit row orthogonality, utilizing the SIPP to identify minimal patterns and guarantee orthogonality with high probability.

Contribution

It introduces new conditions for sign patterns to allow row orthogonality and applies the SIPP to analyze random sign patterns.

Findings

Identified 5x n nowhere zero sign patterns that minimally allow orthogonality.

Established conditions on zero entries guaranteeing the SIPP.

Applied the SIPP to study high-probability orthogonality in random sign patterns.

Abstract

A sign pattern is an array with entries in $\{+,-,0\}$. A 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.