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

## Abstract

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].

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