The eigenstructure of an arbitrary polynomial matrix : Computational aspects
Paul Van Dooren,Patrick Dewilde +1 more
TLDR
In this article, the zero structure, the polar structure, and the left and right null space structure of a polynomial matrix P(λ) have been computed using a new numerical method.About:
This article is published in Linear Algebra and its Applications.The article was published on 1983-04-01 and is currently open access. It has received 144 citations till now. The article focuses on the topics: Matrix polynomial & Polynomial matrix.read more
Citations
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NLEVP: A Collection of Nonlinear Eigenvalue Problems
TL;DR: NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.
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Polynomial roots from companion matrix eigenvalues
Alan Edelman,H. Murakami +1 more
TL;DR: A first- order error analysis of the algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix is derived, which states that the algorithm is backward normwise stable in a sense that must be defined carefully.
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Backward error and condition of polynomial eigenvalue problems
TL;DR: In this article, the authors developed normwise backward errors and condition numbers for the polynomial eigenvalue problem, and showed that solving the QEP by applying the QZ algorithm to a corresponding generalized eigen value problem can be backward unstable.
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Backward Error of Polynomial Eigenproblems Solved by Linearization
TL;DR: The results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of $P and to derive backward error bounds depending only on the norms of the $A_i$ for the companion pencils and for the vector space of pencils recently identified.
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Fiedler Companion Linearizations and the Recovery of Minimal Indices
TL;DR: It is proved that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and it is shown explicitly how to recover the left and right minimal indices and minimal bases of the polynomials from the minimum indices and bases of these linearizations.
References
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Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
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A generalized state-space for singular systems
TL;DR: In this article, a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes.
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An Algorithm for Generalized Matrix Eigenvalue Problems.
Cleve B. Moler,G. W. Stewart +1 more
TL;DR: A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B with particular attention to the degeneracies which result when B is singular.
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Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems
TL;DR: It is shown how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization.