Before revision:  Soc. , in press] and Choudhury [2021 preprint]  .
After revision:  Soc. , in press] and Choudhury [2021 preprint] . Finally using a 1950 result of Gantmacher--Krein, we show that totally negative/non-positive matrices can not be detected by (single) test vectors from orthants other than the open bi-orthant that have coordinates with alternating signs, via the sign non-reversal property or the variation diminishing property .