On the exponent of several classes of oscillatory matrices

Yoram Zarai, Michael Margaliot*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)363-386
Number of pages24
JournalLinear Algebra and Its Applications
StatePublished - 1 Jan 2021


FundersFunder number
US-Israel Binational Science Foundation
United States-Israel Binational Science Foundation2015148
Israel Science Foundation407/19


    • Exponent of oscillatory matrices
    • Planar network
    • Successive elementary factorization
    • Totally nonnegative matrices
    • Totally positive matrices


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