TY - JOUR

T1 - On the exponent of several classes of oscillatory matrices

AU - Zarai, Yoram

AU - Margaliot, Michael

N1 - Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An n×n matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that all minors of Ak are positive. The smallest k for which this holds is called the exponent of the oscillatory matrix A. Gantmacher and Krein showed that the exponent is always smaller than or equal to n−1. An important and nontrivial problem is to determine the exact value of the exponent. Here we use the successive elementary bidiagonal factorization of oscillatory matrices, and its graph-theoretic representation, to derive an explicit expression for the exponent of several classes of oscillatory matrices, and a nontrivial upper-bound on the exponent for several other classes.

AB - Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An n×n matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that all minors of Ak are positive. The smallest k for which this holds is called the exponent of the oscillatory matrix A. Gantmacher and Krein showed that the exponent is always smaller than or equal to n−1. An important and nontrivial problem is to determine the exact value of the exponent. Here we use the successive elementary bidiagonal factorization of oscillatory matrices, and its graph-theoretic representation, to derive an explicit expression for the exponent of several classes of oscillatory matrices, and a nontrivial upper-bound on the exponent for several other classes.

KW - Exponent of oscillatory matrices

KW - Planar network

KW - Successive elementary factorization

KW - Totally nonnegative matrices

KW - Totally positive matrices

UR - http://www.scopus.com/inward/record.url?scp=85091564184&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.09.021

DO - 10.1016/j.laa.2020.09.021

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AN - SCOPUS:85091564184

SN - 0024-3795

VL - 608

SP - 363

EP - 386

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -