On the Exponent of Several Classes of Oscillatory Matrices

DOI10.1016/J.LAA.2020.09.021arXiv1910.10709MaRDI QIDQ6327769

Yoram Zarai, Michael Margaliot

Publication date: 23 October 2019

Abstract: Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An $nimesn$ matrix $A$ is called oscillatory if all its minors are nonnegative and there exists a positive integer $k$ such that all minors of $A^k$ 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.

Mathematics Subject Classification ID

Factorization of matrices (15A23) Determinants, permanents, traces, other special matrix functions (15A15) Special matrices (15B99)

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