Boolean invertible matrices identified from two permutations and their corresponding Haar-type matrices

Abstract

We construct a class R m of m × m boolean invertible matrices whose elements satisfy the following property: when we perform the Hadamard product operation R i ⊙ R j on the set of row vectors { R 1 , … , R m } of an element R ∈ R m we produce either the row R max { i , j } or the zero row. In this paper, we prove that every matrix R ∈ R m is uniquely determined by a pair of permutations of the set { 1 , … , m } . As a by-product of this result we identify Haar-type matrices from a pair of permutations as well, because these matrices emerge from the Gram–Schmidt orthonormalization process of the set of row vectors of R matrices belonging in a certain subclass R 0 ⊂ R m .