Korean J. Math. Vol. 31 No. 2 (2023) pp.145-152
DOI: https://doi.org/10.11568/kjm.2023.31.2.145

On the bounds of the eigenvalues of matrix polynomials

Main Article Content

Wali Mohammad Shah
Zahid Bashir Monga


Let $P(z):=\displaystyle{\sum\limits_{j=0}^n} A_jz^j,~A_j\in \mathbb{C}^{m\times m}, 0\leq j\leq n$ be a matrix polynomial of degree $n,$ such that
$$A_n\geq A_{n-1}\geq \ldots \geq A_0\geq 0,~A_n>0.$$
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 Enestr\"om-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 L\^{e}, Du, Nguy\^{e}n [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].

Article Details


[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. Google Scholar

[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. Google Scholar

[3] G. Dirr and H. K. Wimmer, An Enestrom-Kakeya theorem for hermitian polynomial matrices, IEEE Trans. Automat. Control 52(2007), 2151–2153. Google Scholar

[4] G. Enestrom, Harledning af en allman formel for antallet pensionarer, Ofv. af. Kungl. Vetenskaps-Akademeins Forhandlingen, No. (Stockholm, 1893). Google Scholar

[5] R. B. Gardner and N. K. Govil, Some generalizations of Enestrom-Kakeya theorem, Acta Math. Hungar., 74(1–2) (1997), 125-134. Google Scholar

[6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. Google Scholar

[7] S. Kakeya, On the limits of the roots of an algebraic equations with positive coefficients, Tohoku Math J. 2 (1912-13), 140–142. Google Scholar

[8] C. T. Lˆe, T. H. B. Du, T. D. Nguyˆen, On the location of eigenvalues of matrix polynomials, Oper. Matrices, 13(2019), 937–954. Google Scholar

[9] M. Marden, Geometry of Polynomials, Mathematical Surveys Number 3, Providence, RI: American Mathematical Society (1966). Google Scholar

[10] Q. G. Mohammad, Location of the Zeros of Polynomials, Amer. Math. Monthly, vol.74, No.3, March 1967. Google Scholar