On minimal bases and indices of rational matrices and their linearizations
A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomia...
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| Veröffentlicht in: | Linear algebra and its applications 2021-08, Vol.623, p.14-67 |
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| container_title | Linear algebra and its applications |
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| creator | Amparan, A. Dopico, F.M. Marcaida, S. Zaballa, I. |
| description | A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneering results obtained by Verghese, Van Dooren and Kailath in 1979-1980, which were the first proving results of this type. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices. |
| doi_str_mv | 10.1016/j.laa.2021.01.014 |
| format | Article |
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Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneering results obtained by Verghese, Van Dooren and Kailath in 1979-1980, which were the first proving results of this type. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. 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These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices.</description><subject>[formula omitted]-strong linearizations</subject><subject>Algorithms</subject><subject>Fiedler-like linearizations</subject><subject>Linear algebra</subject><subject>Linearizations</subject><subject>Minimal bases</subject><subject>Minimal indices</subject><subject>Polynomial system matrices</subject><subject>Polynomials</subject><subject>Rational matrices</subject><subject>Strong block minimal bases linearizations</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9UE1LxDAQDaLg-vEDvBU8t06SNmnxJIufLOxFzyEmE0zppmvSFfTXm-56FgZmmHlveO8RckWhokDFTV8NWlcMGK1grvqILGgreUnbRhyTBQCrSy675pScpdQDQC2BLcjLOhQbH_xGD8W7TpgKHWzhg_Umz6Mrop78GPJ1o6e4X86A6QN9LAYfUEf_s4ekC3Li9JDw8q-fk7eH-9flU7laPz4v71al4aKdSgdMUJCWN841zOlOYGfRMueEE4xLDhZkBx2XLW-l40jBSiMbYxzjwgp-Tq4Pf7dx_NxhmlQ_7mKWmBRrBGTTkrOMogeUiWNKEZ3axuwyfisKao5M9SpHpubIFMxVZ87tgYNZ_pfHqJLxGAxaH9FMyo7-H_YvgvJzFA</recordid><startdate>20210815</startdate><enddate>20210815</enddate><creator>Amparan, A.</creator><creator>Dopico, F.M.</creator><creator>Marcaida, S.</creator><creator>Zaballa, I.</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><scope>6I.</scope><scope>AAFTH</scope><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><orcidid>https://orcid.org/0000-0003-2620-7109</orcidid><orcidid>https://orcid.org/0000-0001-7779-8960</orcidid><orcidid>https://orcid.org/0000-0002-6113-9079</orcidid></search><sort><creationdate>20210815</creationdate><title>On minimal bases and indices of rational matrices and their linearizations</title><author>Amparan, A. ; Dopico, F.M. ; Marcaida, S. ; Zaballa, I.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-f026107d35ff52fa96e9ded2ff6f623730d07909378387f3e10d7c75ccf236d63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>[formula omitted]-strong linearizations</topic><topic>Algorithms</topic><topic>Fiedler-like linearizations</topic><topic>Linear algebra</topic><topic>Linearizations</topic><topic>Minimal bases</topic><topic>Minimal indices</topic><topic>Polynomial system matrices</topic><topic>Polynomials</topic><topic>Rational matrices</topic><topic>Strong block minimal bases linearizations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Amparan, A.</creatorcontrib><creatorcontrib>Dopico, F.M.</creatorcontrib><creatorcontrib>Marcaida, S.</creatorcontrib><creatorcontrib>Zaballa, I.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><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>Amparan, A.</au><au>Dopico, F.M.</au><au>Marcaida, S.</au><au>Zaballa, I.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On minimal bases and indices of rational matrices and their linearizations</atitle><jtitle>Linear algebra and its applications</jtitle><date>2021-08-15</date><risdate>2021</risdate><volume>623</volume><spage>14</spage><epage>67</epage><pages>14-67</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. 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| ispartof | Linear algebra and its applications, 2021-08, Vol.623, p.14-67 |
| issn | 0024-3795 1873-1856 |
| language | eng |
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| source | ScienceDirect Journals (5 years ago - present) |
| subjects | [formula omitted]-strong linearizations Algorithms Fiedler-like linearizations Linear algebra Linearizations Minimal bases Minimal indices Polynomial system matrices Polynomials Rational matrices Strong block minimal bases linearizations |
| title | On minimal bases and indices of rational matrices and their linearizations |
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