Sign Patterns of Orthogonal Matrices and the Strong Inner Product Property

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.