Block Kronecker ansatz spaces for matrix polynomials

In this paper, we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangular matrix polynomial. We provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pe...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 542; pp. 118 - 148
Main Authors Faßbender, Heike, Saltenberger, Philip
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.04.2018
American Elsevier Company, Inc
Subjects
Ansatz space
Block Kronecker pencils
Eigenvector recovery
Fiedler pencils
Linear algebra
Linear equations
Linearization
Mathematical analysis
Matrix methods
Matrix polynomials
Pencils
Permutations
Polynomials
Rectangular matrix polynomial
Set theory
Sparsity
Strong linearization
Structure-preserving linearization
Subspaces
Vector spaces
15A03
Fiedler pencils
15A18
Strong linearization
47J10
15A23
15A22
Structure-preserving linearization
Ansatz space
65F15
Rectangular matrix polynomial
Block Kronecker pencils
Eigenvector recovery
Matrix polynomials
Linearization
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Summary:In this paper, we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangular matrix polynomial. We provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pencil therein is a strong linearization regardless whether the matrix polynomial under consideration is regular or singular. These novel “ansatz spaces” cover all block Kronecker pencils as introduced in [6] as a subset and therefore contain all Fiedler pencils modulo permutations. The important case of square matrix polynomials is examined in greater depth. We prove that the intersection of any number of block Kronecker ansatz spaces is never empty and construct large subspaces of block-symmetric matrix pencils among which still almost every pencil is a strong linearization. Moreover, we show that the original ansatz spaces L1 and L2 may essentially be recovered from block Kronecker ansatz spaces via pre- and postmultiplication, respectively, of certain constant matrices.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2017.03.019