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

We construct a class Rm of m×m boolean invertible matrices whose elements satisfy the following property: when we perform the Hadamard product operation Ri⊙Rj on the set of row vectors {R1,…,Rm} of an element R∈Rm we produce either the row Rmax{i,j} or the zero row. In this paper, we prove that every matrix R∈Rm 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 R0⊂Rm.