Integer-Linear Preservers of Binary Matrices

Objectives: Let ℬ𝑛 be the set of 𝑛 × 𝑛 matrices with entries in {0, 1}. This study aims to characterize the ℤ−linear maps from 𝑀𝑛 (ℤ) to 𝑀𝑛 (ℤ) that map the set of binary matrices into itself. Method: We apply vectorization to analyze the representing matrices, evaluate disjoint-support criteria for matrix units, and establish a finite test and module rank enumeration for binary preservers. Findings: We determine that every valid linear map acts as a coordinatecopying transformation where each output entry is zero or a copy of one input coordinate. If the map also preserves ordinary matrix rank, it takes the classical form 𝑇 (𝐴) = 𝑃 𝐴𝑄 or 𝑇 (𝐴) = 𝑃 𝐴𝑇 𝑄 using permutation matrices 𝑃 and 𝑄.Novelty: This work provides a complete classification, exact module rank enumeration, and symmetry analysis for integer-linear binary preservers, demonstrating that symmetry alone is weaker than rank preservation.