On the bounds of eigenvalues of matrix polynomials

被引:0
作者
W. M. Shah
Sooraj Singh
机构
[1] Central University of Kashmir,
来源
The Journal of Analysis | 2023年 / 31卷
关键词
Matrix polynomial; Polynomial eigenvalue problem; Bounds; 15A18; 15A42; 65F15;
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摘要
Let M(z)=Amzm+Am-1zm-1+⋯+A1z+A0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(z)=A_mz^m+A_{m-1}z^{m-1}+\cdots +A_1z+A_0$$\end{document} be a matrix polynomial, whose coefficients Ak∈Cn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k\in {{\mathbb {C}}}^{n\times n}$$\end{document}, ∀k=0,1,…,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \, k=0,1,\ldots , m$$\end{document}, satisfying the following dominant property ‖Am‖>‖Ak‖,∀k=0,1,…,m-1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert A_m\Vert >\Vert A_k\Vert ,\,\forall \, k=0,1,\ldots ,m-1, \end{aligned}$$\end{document}then it is known that all eigenvalues λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} of M(z) locate in the open disk λ<1+‖Am‖‖Am-1‖.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \lambda \right| <1+\Vert A_m\Vert \Vert {A_m}^{-1}\Vert . \end{aligned}$$\end{document}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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页码:821 / 829
页数:8
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