The number of negative entries in a sign pattern allowing orthogonality (Q704842)

The authors show that the following result holds for all \(n\geq 3\). Among the \(n\times n\) real orthogonal matrices without zero entries, the number of negative entries can take any integer value in the interval \([n-1,n^2-n+1]\). Values outside that interval clearly cannot be achieved since they would require the matrix to have two columns in which all entries have the same sign, and such columns cannot be orthogonal.