A matrix A is totally positive (or non‐negative) of order k, denoted TPk (or TNk), if all minors of size ⩽k are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with...
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https://www.riss.kr/link?id=O112659709
2021년
-
0024-6093
1469-2120
SCI;SCIE;SCOPUS
학술저널
981-990 [※수록면이 p5 이하이면, Review, Columns, Editor's Note, Abstract 등일 경우가 있습니다.]
0
상세조회0
다운로드다국어 초록 (Multilingual Abstract)
A matrix A is totally positive (or non‐negative) of order k, denoted TPk (or TNk), if all minors of size ⩽k are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with...
A matrix A is totally positive (or non‐negative) of order k, denoted TPk (or TNk), if all minors of size ⩽k are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non‐reversal property. We do away with the former, and show that A is TPk if and only if every submatrix formed from at most k consecutive rows and columns has the sign non‐reversal property. In fact, this can be strengthened to only consider test vectors in Rk with alternating signs. We also show a similar characterization for all TNk matrices — more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing TP matrices by Gantmacher–Krein (Compos. Math. 4 (1937) 445–476) and P‐matrices by Gale–Nikaido (Math. Ann. 159 (1965) 81–93).
As an application, we study the interval hull I(A,B) of two m×n matrices A=(aij) and B=(bij). This is the collection of C∈Rm×n such that each cij is between aij and bij. Using the sign non‐reversal property, we identify a two‐element subset of I(A,B) that detects the TPk property for all of I(A,B) for arbitrary k⩾1. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non‐negativity (of any order) of an interval hull I(A,B).
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