TY - JOUR

T1 - On descartes’ rule of signs for matrix polynomials

AU - Cameron, Thomas R.

AU - Psarrakos, Panayiotis J.

N1 - Publisher Copyright:
© 2019, Element D.O.O. All rights reserved.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - We present a generalized Descartes’ rule of signs for self-adjoint matrix polynomials whose coefficients are either positive or negative definite, or null. In particular, we conjecture that the number of real positive (negative) eigenvalues of a matrix polynomial is bounded above by the product of the size of the matrix coefficients and the number of definite sign alternations (permanences) between consecutive coefficients. Our main result shows that this generalization holds under the additional assumption that the matrix polynomial is hyperbolic. In addition, we prove individual cases where the matrix polynomial is diagonalizable by congruence, or of degree three or less. The full proof of our conjecture is an open problem; we discuss analytic and algebraic approaches for solving this problem and ultimately, what makes this open problem non-trivial. Finally, we prove generalizations of two famous extensions of Descartes’ rule: If all eigenvalues are real then the bounds in Descartes’ rule are sharp and the number of real positive and negative eigenvalues have the same parity as the associated bounds in Descartes’ rule.

AB - We present a generalized Descartes’ rule of signs for self-adjoint matrix polynomials whose coefficients are either positive or negative definite, or null. In particular, we conjecture that the number of real positive (negative) eigenvalues of a matrix polynomial is bounded above by the product of the size of the matrix coefficients and the number of definite sign alternations (permanences) between consecutive coefficients. Our main result shows that this generalization holds under the additional assumption that the matrix polynomial is hyperbolic. In addition, we prove individual cases where the matrix polynomial is diagonalizable by congruence, or of degree three or less. The full proof of our conjecture is an open problem; we discuss analytic and algebraic approaches for solving this problem and ultimately, what makes this open problem non-trivial. Finally, we prove generalizations of two famous extensions of Descartes’ rule: If all eigenvalues are real then the bounds in Descartes’ rule are sharp and the number of real positive and negative eigenvalues have the same parity as the associated bounds in Descartes’ rule.

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M3 - Article

AN - SCOPUS:85073780214

VL - 13

SP - 643

EP - 652

JO - Operators and Matrices

JF - Operators and Matrices

SN - 1846-3886

IS - 3

ER -