Nakatsukasa, Yuji and Noferini, Vanni and Townsend, Alex (2012) Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach. [MIMS Preprint]
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Abstract
We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, {SIAM J. Matrix Anal. Appl.}, 28 (2006), pp.~9711004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a B\'{e}zout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpration as a space of B\'{e}zout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearization in the important practical case of a Chebyshev basis.
Item Type:  MIMS Preprint 

Uncontrolled Keywords:  matrix polynomials; bivariate polynomials; BÂ´ezoutian; double ansatz space; degreegraded polynomial basis; orthogonal polynomials; conditioning 
Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis 
Depositing User:  Yuji Nakatsukasa 
Date Deposited:  24 Mar 2015 
Last Modified:  08 Nov 2017 18:18 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2276 
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Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach. (deposited 21 Dec 2012)
 Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach. (deposited 24 Mar 2015) [Currently Displayed]
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