On the Exponent of Several Classes of Oscillatory Matrices

TL;DR

This paper investigates the exponent of oscillatory matrices, providing explicit formulas for certain classes and bounds for others using bidiagonal factorizations and graph theory.

Contribution

It introduces a method using bidiagonal factorizations and graph representations to determine the exponent of various classes of oscillatory matrices.

Findings

Explicit exponent formulas for specific classes of oscillatory matrices

Nontrivial upper bounds on the exponent for other classes

Use of bidiagonal factorization and graph theory in analysis

Abstract

Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An $n\times n$ 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.