Intervals of almost totally positive matrices

We consider the class of the totally nonnegative matrices, i.e., the matrices having all their minors nonnegative, and intervals of matrices with respect to the chequerboard partial ordering, which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum. For these intervals in 1982 we stated in this journal the following conjecture: If the left and right endpoints of an interval are nonsingular and totally nonnegative, then all matrices taken from the interval are nonsingular and totally nonnegative. In this paper we show that this conjecture is true if we restrict ourselves to the subset of the almost totally positive matrices.

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