Sign pattern matrices that allow orthogonality (Q1911917)

Two row (column) vectors with entries \(+,-,0\) are said to allow orthogonality, if they are the sign patterns of two orthogonal vectors in \(\mathbb{R}^n\). The author investigates square sign pattern matrices with no zero row and no zero column such that any two distinct rows and any two distinct columns are allowing orthogonality. Any such matrix is called sign potentially orthogonal (SPO). The question is whether every SPO matrix \(A\) allows orthogonality, i.e. \(A\) is the sign pattern matrix of an orthogonal matrix. The answer is negative due to an ad hoc counterexample of a \(5\times 5\) SPO-matrix that does not allow orthogonality. On the other hand sufficient conditions are given for an SPO-matrix to allow orthogonality.