Sign non-reversal property for totally positive matrices and testing total positivity on their interval hull
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We establish a novel characterization of totally positive matrices via a sign non-reversal property. In this we are inspired by the analogous results for P-matrices (matrices with all principal minors positive). Next, the interval hull of two m × n matrices A=(a_ij) and B = (b_ij), denoted by 𝕀(A,B), is the collection of all matrices C such that c_ij=t_ija_ij+(1-t_ij)b_ij for all i,j, where t_ij∈ [0,1]. Using the sign non-reversal property, we identify a finite subset of 𝕀(A,B) that detects the total positivity of all of 𝕀(A,B). This provides a test for an entire class of matrices simultaneously to be totally positive. We also establish analogous results for other classes of matrices: almost P-matrices, N-matrices and (minimally) semipositive matrices.
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