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
SN - 1846-3886
VL - 13
SP - 643
EP - 652
JO - Operators and Matrices
JF - Operators and Matrices
IS - 3
ER -