{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/54268"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/54268","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Zero-pole interpolation of nonregular rational matrix functions","abstract":"In this thesis the right and left pole structure of a not necessarily regular rational matrix function W is described in terms of pairs of matrices-right and left pole pairs. The concept of orthogonality in R<sup>n</sup> is investigated. Using this concept, the right and left zero structure of a rational matrix function W is described in terms of pairs and triples of matrices-right and left null pairs and right and left kernel triples. The definition of a spectral triple of a regular rational matrix function over a subset σ of C is extended to the nonregular case. Given a rational matrix function W and a subset σ of C, the left null-pole subspace of W over σ is described in terms of a left kernel triple and a left σ-spectral triple for W. A sufficient condition for the minimality of McMillan degree of a rational matrix function H which is right equivalent to W on σ, that is a rational matrix function H of the same size and with the same left null-pole subspace over σ as W, is developed. An algorithm for constructing a rational matrix function W with a left kernel triple (A<sub>κ</sub> B<sub>κ</sub> D<sub>κ</sub>) and left null and right pole pairs over σ⊂C (A<sub>ζ</sub>, B<sub>ζ</sub>) and (C<sub>π</sub>, A<sub>π</sub>), respectively, from a regular rational matrix function with left null and right pole pairs over σ (A<sub>ζ</sub>, B<sub>ζ</sub>) and (C<sub>π</sub>, A<sub>π</sub>) is described. Finally, a necessary and sufficient condition for existence of a rational matrix function W with a given left kernel triple and a given left spectral triple over a subset σ of C is established.","abstract_html":"In this thesis the right and left pole structure of a not necessarily regular rational matrix function W is described in terms of pairs of matrices-right and left pole pairs. The concept of orthogonality in R&lt;sup&gt;n&lt;/sup&gt; is investigated. Using this concept, the right and left zero structure of a rational matrix function W is described in terms of pairs and triples of matrices-right and left null pairs and right and left kernel triples. The definition of a spectral triple of a regular rational matrix function over a subset σ of C is extended to the nonregular case. Given a rational matrix function W and a subset σ of C, the left null-pole subspace of W over σ is described in terms of a left kernel triple and a left σ-spectral triple for W. A sufficient condition for the minimality of McMillan degree of a rational matrix function H which is right equivalent to W on σ, that is a rational matrix function H of the same size and with the same left null-pole subspace over σ as W, is developed. An algorithm for constructing a rational matrix function W with a left kernel triple (A&lt;sub&gt;κ&lt;/sub&gt; B&lt;sub&gt;κ&lt;/sub&gt; D&lt;sub&gt;κ&lt;/sub&gt;) and left null and right pole pairs over σ⊂C (A&lt;sub&gt;ζ&lt;/sub&gt;, B&lt;sub&gt;ζ&lt;/sub&gt;) and (C&lt;sub&gt;π&lt;/sub&gt;, A&lt;sub&gt;π&lt;/sub&gt;), respectively, from a regular rational matrix function with left null and right pole pairs over σ (A&lt;sub&gt;ζ&lt;/sub&gt;, B&lt;sub&gt;ζ&lt;/sub&gt;) and (C&lt;sub&gt;π&lt;/sub&gt;, A&lt;sub&gt;π&lt;/sub&gt;) is described. Finally, a necessary and sufficient condition for existence of a rational matrix function W with a given left kernel triple and a given left spectral triple over a subset σ of C is established.","abstract_has_math":false,"creators":["Rakowski, Marek"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Ball, Joseph A."],"committee_members":["Hannsgen, Kenneth B.","Thomson, James E.","Wheeler, Robert","Brown, Ezra A."],"year":1989,"date_issued":"1989","date_published":"1989","updated_at":"2026-07-22T22:20:32Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/54268","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Ball, Joseph A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Hannsgen, Kenneth B.","Thomson, James E.","Wheeler, Robert","Brown, Ezra A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Rakowski, Marek"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-07-09T20:43:29Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-07-09T20:43:29Z"]},{"key":"dc:date.issued","label":"Date","values":["1989"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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Using this concept, the right and left zero structure of a rational matrix function W is described in terms of pairs and triples of matrices-right and left null pairs and right and left kernel triples. The definition of a spectral triple of a regular rational matrix function over a subset σ of C is extended to the nonregular case. Given a rational matrix function W and a subset σ of C, the left null-pole subspace of W over σ is described in terms of a left kernel triple and a left σ-spectral triple for W. A sufficient condition for the minimality of McMillan degree of a rational matrix function H which is right equivalent to W on σ, that is a rational matrix function H of the same size and with the same left null-pole subspace over σ as W, is developed. An algorithm for constructing a rational matrix function W with a left kernel triple (A<sub>κ</sub> B<sub>κ</sub> D<sub>κ</sub>) and left null and right pole pairs over σ⊂C (A<sub>ζ</sub>, B<sub>ζ</sub>) and (C<sub>π</sub>, A<sub>π</sub>), respectively, from a regular rational matrix function with left null and right pole pairs over σ (A<sub>ζ</sub>, B<sub>ζ</sub>) and (C<sub>π</sub>, A<sub>π</sub>) is described. Finally, a necessary and sufficient condition for existence of a rational matrix function W with a given left kernel triple and a given left spectral triple over a subset σ of C is established."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Zero-pole interpolation of nonregular rational matrix functions"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Ball, Joseph A."],"dc:contributor.committeemember":["Hannsgen, Kenneth B.","Thomson, James E.","Wheeler, Robert","Brown, Ezra A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Rakowski, Marek"],"dc:date.accessioned":["2015-07-09T20:43:29Z"],"dc:date.available":["2015-07-09T20:43:29Z"],"dc:date.issued":["1989"],"dc:description.abstract":["In this thesis the right and left pole structure of a not necessarily regular rational matrix function W is described in terms of pairs of matrices-right and left pole pairs. The concept of orthogonality in R<sup>n</sup> is investigated. Using this concept, the right and left zero structure of a rational matrix function W is described in terms of pairs and triples of matrices-right and left null pairs and right and left kernel triples. The definition of a spectral triple of a regular rational matrix function over a subset σ of C is extended to the nonregular case. Given a rational matrix function W and a subset σ of C, the left null-pole subspace of W over σ is described in terms of a left kernel triple and a left σ-spectral triple for W. A sufficient condition for the minimality of McMillan degree of a rational matrix function H which is right equivalent to W on σ, that is a rational matrix function H of the same size and with the same left null-pole subspace over σ as W, is developed. An algorithm for constructing a rational matrix function W with a left kernel triple (A<sub>κ</sub> B<sub>κ</sub> D<sub>κ</sub>) and left null and right pole pairs over σ⊂C (A<sub>ζ</sub>, B<sub>ζ</sub>) and (C<sub>π</sub>, A<sub>π</sub>), respectively, from a regular rational matrix function with left null and right pole pairs over σ (A<sub>ζ</sub>, B<sub>ζ</sub>) and (C<sub>π</sub>, A<sub>π</sub>) is described. Finally, a necessary and sufficient condition for existence of a rational matrix function W with a given left kernel triple and a given left spectral triple over a subset σ of C is established."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/54268"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Zero-pole interpolation of nonregular rational matrix functions"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:32Z"}