Abstract
AbstractIt is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.
Highlights
An m-by-n matrix is totally positive, or TP (TN), if all its minors are positive
Since scaling of any line in a TP matrix leaves it TP, it follows readily that a line may be inserted with a speci ed positive value in any particular position
The two speci ed entries must themselves be consistent with total positivity
Summary
An m-by-n matrix is totally positive (totally nonnegative), or TP (TN), if all its minors are positive (nonnegative). It has long been known that any TP matrix may be bordered (either on the side, top, or bottom - see page 185 of [3] for an indicative diagram and explanation) so as to remain TP This is a convenient way to generate TP matrices of arbitrary size. Much more subtly, it was shown in [4] that between any two consecutive lines (rows or columns) of a TP matrix, a line may be inserted so as to remain TP. The two speci ed entries must themselves be consistent with total positivity (i.e. the resulting partial matrix must be partial TP [3], which we assume) This is of considerable interest in TP completion problems.
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