Affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases
The main aim of this paper is two-fold. First, to define and analyze two affine spaces of matrix pencils associated with a realization of an n×n rational matrix G(λ) and show that almost all of these pencils are strong linearizations of G(λ). Second, to describe the recovery of eigenvectors, minimal...
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| Veröffentlicht in: | Linear algebra and its applications 2019-05, Vol.569, p.335-368 |
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| creator | Das, Ranjan Kumar Alam, Rafikul |
| description | The main aim of this paper is two-fold. First, to define and analyze two affine spaces of matrix pencils associated with a realization of an n×n rational matrix G(λ) and show that almost all of these pencils are strong linearizations of G(λ). Second, to describe the recovery of eigenvectors, minimal bases and minimal indices of G(λ) from those of the strong linearizations of G(λ). The affine spaces of pencils are constructed from the vector spaces of linearizations of matrix polynomials defined and analyzed by Mackey et al. in [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971–1004]. We show that eigenvectors, minimal bases and minimal indices of G(λ) can be easily recovered from those of the strong linearizations of G(λ). In particular, we construct a symmetric strong linearization of G(λ) when G(λ) is regular and symmetric. We also describe the recovery of pole-zero structure of G(λ) at infinity from the pole-zero structure at infinity of a strong linearization of G(λ). |
| doi_str_mv | 10.1016/j.laa.2019.02.001 |
| format | Article |
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First, to define and analyze two affine spaces of matrix pencils associated with a realization of an n×n rational matrix G(λ) and show that almost all of these pencils are strong linearizations of G(λ). Second, to describe the recovery of eigenvectors, minimal bases and minimal indices of G(λ) from those of the strong linearizations of G(λ). The affine spaces of pencils are constructed from the vector spaces of linearizations of matrix polynomials defined and analyzed by Mackey et al. in [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971–1004]. We show that eigenvectors, minimal bases and minimal indices of G(λ) can be easily recovered from those of the strong linearizations of G(λ). In particular, we construct a symmetric strong linearization of G(λ) when G(λ) is regular and symmetric. 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We also describe the recovery of pole-zero structure of G(λ) at infinity from the pole-zero structure at infinity of a strong linearization of G(λ).</description><subject>Construction</subject><subject>Eigenvalue</subject><subject>Eigenvector</subject><subject>Eigenvectors</subject><subject>Infinity</subject><subject>Linear algebra</subject><subject>Linearization</subject><subject>Mathematical analysis</subject><subject>Matrix algebra</subject><subject>Matrix methods</subject><subject>Matrix pencil</subject><subject>Matrix polynomial</subject><subject>Minimal realization</subject><subject>Pencils</subject><subject>Polynomials</subject><subject>Rational matrices</subject><subject>Recovery</subject><subject>Strong linearization</subject><subject>System matrix</subject><subject>Vector spaces</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLAzEURoMoWKs_wF3A9Yz3JvPEVRFfUHCj65AmNzWlndRkLNRfb2pdu0oI54SPw9g1QomAze2qXGtdCsC-BFEC4AmbYNfKAru6OWUTAFEVsu3rc3aR0goAqhbEhIWZc34gnrbaUOLB8TTGMCz5Or_q6L_16MOQuAuRx9-7XvONHqM_4HqwfPwgHsmEHcX9wSe_pGFHZgzxCGz84DfZWuhE6ZKdOb1OdPV3Ttn748Pb_XMxf316uZ_NCyObbiysE2h62za9taQX0mhdO5RGWtG0XSs629ZSQt1WWrsFAfQSFw2izLxrDMopuzn-u43h84vSqFbhK-bxSQmBEqu-r5pM4ZEyMaQUyaltzFvjXiGoQ1e1UrmrOnRVIFTump27o0N5_s5TVMl4GgxZnzOMygb_j_0DZTyCBg</recordid><startdate>20190515</startdate><enddate>20190515</enddate><creator>Das, Ranjan Kumar</creator><creator>Alam, Rafikul</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20190515</creationdate><title>Affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases</title><author>Das, Ranjan Kumar ; Alam, Rafikul</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-df21c9d769ddeab3caa5f13c3d2678728d75330574aafbe00931b61139ddf6c13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Construction</topic><topic>Eigenvalue</topic><topic>Eigenvector</topic><topic>Eigenvectors</topic><topic>Infinity</topic><topic>Linear algebra</topic><topic>Linearization</topic><topic>Mathematical analysis</topic><topic>Matrix algebra</topic><topic>Matrix methods</topic><topic>Matrix pencil</topic><topic>Matrix polynomial</topic><topic>Minimal realization</topic><topic>Pencils</topic><topic>Polynomials</topic><topic>Rational matrices</topic><topic>Recovery</topic><topic>Strong linearization</topic><topic>System matrix</topic><topic>Vector spaces</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Das, Ranjan Kumar</creatorcontrib><creatorcontrib>Alam, Rafikul</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Das, Ranjan Kumar</au><au>Alam, Rafikul</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases</atitle><jtitle>Linear algebra and its applications</jtitle><date>2019-05-15</date><risdate>2019</risdate><volume>569</volume><spage>335</spage><epage>368</epage><pages>335-368</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>The main aim of this paper is two-fold. First, to define and analyze two affine spaces of matrix pencils associated with a realization of an n×n rational matrix G(λ) and show that almost all of these pencils are strong linearizations of G(λ). Second, to describe the recovery of eigenvectors, minimal bases and minimal indices of G(λ) from those of the strong linearizations of G(λ). The affine spaces of pencils are constructed from the vector spaces of linearizations of matrix polynomials defined and analyzed by Mackey et al. in [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971–1004]. We show that eigenvectors, minimal bases and minimal indices of G(λ) can be easily recovered from those of the strong linearizations of G(λ). In particular, we construct a symmetric strong linearization of G(λ) when G(λ) is regular and symmetric. 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| subjects | Construction Eigenvalue Eigenvector Eigenvectors Infinity Linear algebra Linearization Mathematical analysis Matrix algebra Matrix methods Matrix pencil Matrix polynomial Minimal realization Pencils Polynomials Rational matrices Recovery Strong linearization System matrix Vector spaces |
| title | Affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases |
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