Abstract
Let \(M(z)=A_mz^m+A_{m-1}z^{m-1}+\cdots +A_1z+A_0\) be a matrix polynomial, whose coefficients \(A_k\in {{\mathbb {C}}}^{n\times n}\), \(\forall \, k=0,1,\ldots , m\), satisfying the following dominant property
then it is known that all eigenvalues \(\lambda\) of M(z) locate in the open disk
In this paper, among other things, we prove some refinements of this result, which in particular provide refinements of some results concerning the distribution of zeros of polynomials in the complex plane.
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Acknowledgements
We are highly thankful to the referee for his/her invaluable suggestions.
Funding
The first author acknowledges the financial support given by the Science and Engineering Research Board, Govt of India under Mathematical Research Impact-Centric Sport(MATRICS) Scheme vide SERB Sanction order No: F : MTR/2017/000508, Dated 28-05-2018.
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Communicated by S Ponnusamy.
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Shah, W.M., Singh, S. On the bounds of eigenvalues of matrix polynomials. J Anal 31, 821–829 (2023). https://doi.org/10.1007/s41478-022-00481-3
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DOI: https://doi.org/10.1007/s41478-022-00481-3