Determinantal mates (Q2908991)

Let \(\Omega_n\) denote the set of all \(n\times n\) doubly stochastic matrices. Given any real valued function \(f\) on \(\Omega_n\) two distinct matrices \(A\) and \(B\) in \(\Omega_n\) are called \(f\)-mates if the function \(f\) is constant on the segment \(tA+(1-t)B\), \(0\leq t\leq 1\). The authors choose \(f\) to be the determinant function and give necessary and sufficient condition for a symmetric matrix \(A\in\Omega_n\) and a perturbation matrix \(A+E\in\Omega_n\) to be determinantal mates.