A new family of companion forms of polynomial matrices

By E. Antoniou, S. Vologiannidis. 2004

149 citations recorded by the source corpus.

In this paper a new family of companion forms associated to a regular polynomial matrix is presented. Similar results have been presented in a recent paper by M. Fiedler, where the scalar case is considered. It is shown that the new family of companion forms preserves both the finite and infinite elementary divisors structure of the original polynomial matrix, thus all its members can be seen as linearizations of the corresponding polynomial matrix. Furthermore, for the special class of self-adjoint polynomial matrices a particular member is shown to be self-adjoint itself.

Subfields: Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/af0b11403e558d21cd734bb69bac8800754e34ec

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In Atomic and Molecular Physics, and Optics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Jordan structures of alternating matrix polynomials (2010, methodology): “However, this result about SP (λ) can be significantly strengthened, using simpler and more direct arguments than in [1].”
Spectral equivalence of matrix polynomials and the index sum theorem (2014, background): “Therefore the development of structure-preserving algorithms for computing eigenvalues of structured matrix polynomials has become a hot area of research in numerical linear algebra in recent years and, as a consequence, considerable effort has been invested in devising linearizations of matrix poly”
LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES (2009, background): “The results in this paper on linearizations from L1(P ) and L2(P ) for singular polynomials have been extended to include the companion-like pencils that were introduced in [4] as linearizations for regular polynomials.”
Smith Forms of Palindromic Matrix Polynomials (2011, unknown): “These pencils were introduced and analyzed in [22], extending and simplifying results in [1].”
Linearization of regular matrix polynomials (2008, background): “But see also some more recent alternatives to the classical companion forms developed in [2] and [3].”
Fiedler companion linearizations for rectangular matrix polynomials (2012, methodology): “This is the family of Fiedler pencils, which was originally introduced by Fiedler for scalar polynomials in [14], generalized to regular matrix polynomials over C in [2], and then extended and further analyzed in [11] for both regular and singular square matrix polynomials over an arbitrary field F.”
A framework for analyzing nonlinear eigenproblems and parametrized linear systems (2011, background): “3 ( linearizations of Antoniou and Vologiannidis ) With the notation M 0 = diag ( −I n ( ℓ−1 ) , A 0 ) , M ℓ = diag ( A ℓ , I ( ℓ−1 ) n ) , M j = diag I n ( j−1 ) , −A ℓ−j I n I n 0 , I ( ℓ−j−1 ) , j = 1 : ℓ − 1 and any four ordered sets of indices I k = {i k , 1 , i k , 2 , . . . , i k , n k } , k ”