# Sign Patterns of Orthogonal Matrices and the Strong Inner Product Property

## TL;DR

This paper introduces the strong inner product property, a new condition for constructing sign patterns of orthogonal matrices, and provides algorithms for verifying this property, revealing new combinatorial insights.

## Contribution

The paper presents the strong inner product property, new algorithmic techniques for verification, and constructs infinite families of sign patterns for row orthogonal matrices.

## Key findings

- Infinite families of sign patterns allowing row orthogonality
- Algorithmic methods for verifying the strong inner product property
- Insights into the combinatorial structure of orthogonal matrices

## Abstract

A new condition, the strong inner product property, is introduced and used to construct sign patterns of row orthogonal matrices. Using this property, infinite families of sign patterns allowing row orthogonality are found. These provide insight into the underlying combinatorial structure of row orthogonal matrices. Algorithmic techniques for verifying that a matrix has the strong inner product property are also presented. These techniques lead to a generalization of the strong inner product property and can be easily implemented using various software.