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TP and TN Completability of Border Patterns

Chang, Haoge
Abstract
A matrix is called totally nonnegative (TN) if the determinant of every square submatrix is nonnegative and totally positive (TP) if the determinant of every square submatrix is positive. The TP (TN) completion problem asks which partial matrices have a TP (TN) completion. In this paper, a new class of TP- and TN- completable patterns, the border patterns, is identified. This answers an unpublished question about TP-completable patterns that has been outstanding for some time and is the first case of completable patterns with all entries on the border specified. In the process, a new tool is developed: TP line insertion in the second or penultimate line when the first and last entries of the line are specified. Prior results about single unspecified entries are used and generalized.
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2017-04-01
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Undergraduate Honors Theses
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Johnson, Charles R.
Campbell, Donald E.
Yu, Gexin
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Mathematics
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https://scholarworks.wm.edu/handle/internal/10399
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