Korean Journal of Mathematics

  • 제31권2호
  • /
  • Pages.145-152
  • /
  • 2023
  • /
  • 1976-8605(pISSN)
  • /
  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE BOUNDS OF THE EIGENVALUES OF MATRIX POLYNOMIALS

  • 투고 : 2023.01.28
  • 심사 : 2023.04.14
  • 발행 : 2023.06.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let $P(z):=\sum\limits^{n}_{j=0}A_jz^j$, Aj ∈ ℂm×m, 0 ≤ j ≤ n be a matrix polynomial of degree n, such that An ≥ An-1 ≥ . . . ≥ A0 ≥ 0, An > 0. Then the eigenvalues of P(z) lie in the closed unit disk. This theorem proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007), 2151-2153] is infact a matrix extension of a famous and elegant result on the distribution of zeros of polynomials known as Eneström-Kakeya theorem. In this paper, we prove a more general result which inter alia includes the above result as a special case. We also prove an improvement of a result due to Lê, Du, Nguyên [Oper. Matrices, 13(2019), 937-954] besides a matrix extention of a result proved by Mohammad [Amer. Math. Monthly, vol.74, No.3, March 1967].

키워드

참고문헌

  1. L. V. Ahlfors, Complex analysis: an introduction to the theory of analytic functions of one complex variable, 3rd ed., McGraw-Hill Book Co., New York, 1978.
  2. A. Aziz and Q. G. Mohammad, On the zeros of a certain class of polynomials and related analytic functions, J. Math. Anal. Appl. 75 (1980), 495-502. https://doi.org/10.1016/0022-247X(80)90097-9
  3. G. Dirr and H. K. Wimmer, An Enestrom-Kakeya theorem for hermitian polynomial matrices, IEEE Trans. Automat. Control 52 (2007), 2151-2153. https://doi.org/10.1109/TAC.2007.908340
  4. G. Enestrom, Harledning af en allman formel for antalet pensionarer, Ofv. af. Kungl. Vetenskaps-Akademeins Forhandlingen, No. (Stockholm, 1893).
  5. R. B. Gardner and N. K. Govil, Some generalizations of Enestrom-Kakeya theorem, Acta Math. Hungar., 74(1-2) (1997), 125-134. https://doi.org/10.1007/BF02697881
  6. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.
  7. S. Kakeya, On the limits of the roots of an algebraic equations with positive coefficients, Tohoku Math J. 2 (1912-13), 140-142.
  8. C. T. Le, T. H. B. Du, T. D. Nguyen, On the location of eigenvalues of matrix polynomials, Oper. Matrices, 13(2019), 937-954. https://doi.org/10.7153/oam-2019-13-66
  9. M. Marden, Geometry of Polynomials, Mathematical Surveys Number 3, Providence, RI: American Mathematical Society (1966).
  10. Q. G. Mohammad, Location of the Zeros of Polynomials, Amer. Math. Monthly, vol.74, No.3, March 1967.