Gram mates, sign changes in singular values, and isomorphism

TL;DR

This paper investigates the properties and classifications of Gram mates, special (0,1) matrices with equal Gram matrices, focusing on sign changes in singular values and conditions for isomorphism and differences of low rank.

Contribution

It provides a comprehensive characterization of Gram mates, especially those related by sign changes in singular values and with low-rank differences, including new construction tools.

Findings

Characterized Gram mates related by sign changes in singular values.

Classified Gram mates with difference rank at most 2.

Provided conditions for non-isomorphic Gram mates with rank 1 difference.

Abstract

We study distinct $(0,1)$ matrices $A$ and $B$, called \textit{Gram mates}, such that $AA^T=BB^T$ and $A^TA=B^TB$. We characterize Gram mates where one can be obtained from the other by changing signs of some positive singular values. We classify Gram mates such that the rank of their difference is at most $2$. Among such Gram mates, we further produce equivalent conditions in order that one is obtained from the other by changing signs of at most $2$ positive singular values. Moreover, we provide some tools for constructing Gram mates where the rank of their difference is more than $2$. Finally, we characterize non-isomorphic Gram mates whose difference is of rank $1$ with some extra conditions.