On vector spaces of linearizations for matrix polynomials in orthogonal bases
| Autor: | Heike Faßbender, Philip Saltenberger |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory 010102 general mathematics Numerical Analysis (math.NA) Mathematics - Rings and Algebras 010103 numerical & computational mathematics 65F15 15A03 15A18 15A22 01 natural sciences Orthogonal basis Matrix polynomial Combinatorics Rings and Algebras (math.RA) Linearization Singular matrix FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Numerical Analysis Geometry and Topology 0101 mathematics Algebraic number Vector space Mathematics |
| Zdroj: | Linear Algebra and its Applications. 525:59-83 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2017.03.017 |
| Popis: | Regular and singular matrix polynomials P ( λ ) = ∑ i = 0 k P i ϕ i ( λ ) , P i ∈ R n × n given in an orthogonal basis ϕ 0 ( λ ) , ϕ 1 ( λ ) , … , ϕ k ( λ ) are considered. Following the ideas in [9] , the vector spaces, called M 1 ( P ) , M 2 ( P ) and DM ( P ) , of potential linearizations for P ( λ ) are analyzed. All pencils in M 1 ( P ) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M 1 ( P ) is a (strong) linearization of P ( λ ) are given. The equivalence of some of them to the Z-rank-condition [9] is pointed out. Results on the vector space dimensions, the genericity of linearizations in M 1 ( P ) and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Moreover, an extension of these results to degree-graded bases is presented. Throughout the paper, structural resemblances between the matrix pencils in L 1 , i.e. the results obtained in [9] , and their generalized versions are pointed out. |
| Databáze: | OpenAIRE |
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