Sign pattern matrices that allow orthogonality Original Research Article

A pair of sign pattern row vectors (or column vectors) allows orthogonality if the two vectors are the sign patterns for two real orthogonal row vectors (respectively, column vectors). A square sign pattern matrix that does not have a zero row or zero column is sign potentially orthogonal (SPO) if every pair of rows and every pair of columns allows orthogonality. It has been conjectured that every SPO matrix A allows orthogonality; that is, there is a real orthogonal matrix H whose sign pattern equals A. A counterexample to this conjecture is presented, and sufficient conditions are given for a ± SPO matrix to allow orthogonality.