Vector Spaces of Linearizations for Matrix Polynomials

By D. Steven Mackey, Niloufer Mackey, Christian Mehl, Volker Mehrmann, Mims EPrint. SIAM Journal on Matrix Analysis and Applications, 2006

362 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/4cf6e4870dfe7f904b693660c6dd0eb927e2ad40

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Citations in the corpus

Linearization of matrix polynomials expressed in polynomial bases (2008, background): “In the more suggestive notation of [21] this equation reads: 2 ( Φ (λ)⊗ I ) (λC1 − C0) = kn−1en ⊗P(λ) where ΦT (λ) = [φ0(λ), φ1(λ), .”
Balanced truncation model reduction of second-order systems (2008, background): “Note that instead of (4) one can also take other first-order systems that keep the structure in E and A for structured M, D and K, see [3, 4, 9].”
Chebyshev interpolation for nonlinear eigenvalue problems (2012, background): “This transformation is not at all unique [23].”
A numerical method for polynomial eigenvalue problems using contour integral (2010, background): “Polynomial eigenvalue problems are typically solved by linearization [6, 13, 15], which promotes the k-th order n × n matrix polynomial into the larger kn × kn linear eigenvalue problem.”
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations (2006, background): “In this paper we show that the pencil spaces L 1 (P ) and L 2 (P ), developed in [27] by generalizing the first and second companion forms, are rich enough to include subspaces of pencils that reflect palindromic, even, or odd structure of a matrix polynomial P .”
LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES (2009, background): “It was stated and proved in [24], but we include it here for completeness.”
Fiedler companion linearizations for rectangular matrix polynomials (2012, background): “, the classes related to Fiedler pencils considered in [2, 5, 12, 34], or the vector spaces of linearizations introduced in [27].”
Linearization of regular matrix polynomials (2008, background): “3Although “strong linearization” is recognized in the recent work of [12], the hypothesis that the leading coefficient is nonzero is maintained.”
Möbius transformations of matrix polynomials (2015, background): “The reversal of a (matrix) polynomial is an important notion used to define infinite elementary divisors and strong linearizations of matrix polynomials [29, 41, 45].”
Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods (2004, methodology): “It has been shown in [56] that in order to properly reflect the infinite eigenvalues (which occur if the leading coefficient Mk is singular) one should consider matrix pencils L(λ) that satisfy one or both conditions in (10).”
Spectral equivalence of matrix polynomials and the index sum theorem (2014, background): “Recent work in [8, 9, 11] provides answers to these questions for the special cases of a singular polynomial P and its linearizations in the pencil spaces L1(P ) and L2(P ) introduced in [39], as well as for its Frobenius and Fiedler companion linearizations introduced in [2].”