ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Total Positivity

Tue 16:00–16:30, Auditorium Max Weber
A criterion of total nonnegativity of generalized Hurwitz matrices and root locations of polynomials
Mikhail Tyaglov (Shanghai Jiao Tong University)
Joint work with Olga Kushel

For a given real polynomial % $$ p(z)=a_0z^n+a_1z^{n-1}+\cdots+a_n, \qquad a_0>0, $$ % and an integer $M$, $2\leqslant M\leqslant n$, the infinite matrix $H^{(M)}_{\infty}=(a_{Mj-i})_{i,j\in\mathbb{Z}}$ is called generalized Hurwitz matrix, and for $M=2$ the matrix $H^{(2)}_{\infty}$ is the standard infinite Hurwitz matrix. It is known [1,3] that the total positivity of the matrix $H^{(2)}_{\infty}$ is equivalent to the positivity of the first $n$ leading principal minors of $H^{(2)}_{\infty}$. In [2] there were found finitely many sufficient conditions for the matrix $H^{(M)}_{\infty}$ to be totally positive. In this talk, we show that positivity of finitely many certain minors of the generalized Hurwitz matrix $H^{(M)}_{\infty}$, $2\leqslant M\leqslant n$, is necessary and sufficient for total positivity of the matrix $H^{(M)}_{\infty}$.

We also present some applications of totally positive generalized Hurwitz matrices to the root location of polynomials.

  1. B. A. Asner, Jr. On the total nonnegativity of the Hurwitz matrix. SIAM J. Appl. Math., 18:407–414, 1970.
  2. T. N. T. Goodman and Q. Sun. Total positivity and refinable functions with general dilation. Applied and Computational Harmonic Analysis, 16:69–89, 2004.
  3. J. H. B. Kemperman. A Hurwitz matrix is totally positive. SIAM J. Math. Anal., 13(2):331–341, 1982.