{"id":{"repo_id":"aalto","oai_identifier":"oai:aaltodoc.aalto.fi:123456789/135593"},"canonical_url":"https://search.dev.ndltd.org/etd/aalto/oai:aaltodoc.aalto.fi:123456789/135593","repository":{"repo_id":"aalto","name":"Aalto University","base_url":"https://aaltodoc.aalto.fi/server/oai/request"},"display":{"title":"Eigenstructure of polynomial and rational matrices: perturbation theory and nearness problems","abstract":"Polynomial and rational matrices are important algebraic objects that have many engineering applications, most notably in control theory. As with more classical matrices over the field of real or complex numbers, many of the properties of polynomial and rational matrices can be described in terms of their eigenstructure. This thesis considers questions related to the eigenstructure of polynomial and rational matrices. In particular, we study how variations in the entries affect the eigenstructure, and vice versa. These questions generally fall under either perturbation theory or matrix nearness problems, and are important in, for example, understanding how much error there can be in a numerical computation of the eigenstructure. This thesis begins with an overview on the theory of polynomial and rational matrices and their eigenstructures and continues by providing new scientific results in the form of four attached articles. In the first article, we derive new results for the conditioning of zeros of rational matrices. This is achieved by using the transfer function representation for rational matrices together with Rosenbrock's theorem, which relates the eigenstructure of a rational matrix with that of the associated polynomial system matrix. These results have significance in control theory, where rational matrices appear as transfer function matrices in a natural way. The remaining three articles deal with nearness problems associated with the eigenstructure of polynomial matrices. These include the problem of finding the distance to singularity and the problem of finding the distance to stability. For these problems, we use a novel approach that relies on Riemannian optimization. This approach leads to computational algorithms that often outperform the competitors in terms of the running time as well as the quality of the output.","abstract_html":"Polynomial and rational matrices are important algebraic objects that have many engineering applications, most notably in control theory. As with more classical matrices over the field of real or complex numbers, many of the properties of polynomial and rational matrices can be described in terms of their eigenstructure. This thesis considers questions related to the eigenstructure of polynomial and rational matrices. In particular, we study how variations in the entries affect the eigenstructure, and vice versa. These questions generally fall under either perturbation theory or matrix nearness problems, and are important in, for example, understanding how much error there can be in a numerical computation of the eigenstructure. This thesis begins with an overview on the theory of polynomial and rational matrices and their eigenstructures and continues by providing new scientific results in the form of four attached articles. In the first article, we derive new results for the conditioning of zeros of rational matrices. This is achieved by using the transfer function representation for rational matrices together with Rosenbrock's theorem, which relates the eigenstructure of a rational matrix with that of the associated polynomial system matrix. These results have significance in control theory, where rational matrices appear as transfer function matrices in a natural way. The remaining three articles deal with nearness problems associated with the eigenstructure of polynomial matrices. These include the problem of finding the distance to singularity and the problem of finding the distance to stability. For these problems, we use a novel approach that relies on Riemannian optimization. This approach leads to computational algorithms that often outperform the competitors in terms of the running time as well as the quality of the output.","abstract_has_math":false,"creators":["Nyman, Lauri"],"institution":"Aalto University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Matematiikan ja systeemianalyysin laitos","school":null,"contributors":["Aalto-yliopisto","Aalto University"],"advisors":["Noferini, Vanni, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025","date_published":"2025","updated_at":"2026-08-21T22:21:56Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aaltodoc.aalto.fi/handle/123456789/135593","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"source_record":{"url":"https://aaltodoc.aalto.fi/server/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Aaaltodoc.aalto.fi%3A123456789%2F135593","prefix":"dim"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Aalto-yliopisto","Aalto University"]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Noferini, Vanni, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland"]},{"key":"dc:contributor.department","label":"Department","values":["Matematiikan ja systeemianalyysin laitos","Department of Mathematics and Systems Analysis"]},{"key":"dc:contributor.supervisor","label":"Supervisor","values":["Noferini, Vanni, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland"]},{"key":"dc:creator","label":"Author","values":["Nyman, Lauri"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-05-20T09:00:35Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-05-20T09:00:35Z"]},{"key":"dc:date.issued","label":"Date","values":["2025"]},{"key":"dc:publisher","label":"Institution","values":["Aalto University","Aalto-yliopisto"]},{"key":"dc:type","label":"Dc Type","values":["G5 Artikkeliväitöskirja"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["text"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://aaltodoc.aalto.fi/handle/123456789/135593"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Polynomial and rational matrices are important algebraic objects that have many engineering applications, most notably in control theory. As with more classical matrices over the field of real or complex numbers, many of the properties of polynomial and rational matrices can be described in terms of their eigenstructure. This thesis considers questions related to the eigenstructure of polynomial and rational matrices. In particular, we study how variations in the entries affect the eigenstructure, and vice versa. These questions generally fall under either perturbation theory or matrix nearness problems, and are important in, for example, understanding how much error there can be in a numerical computation of the eigenstructure. This thesis begins with an overview on the theory of polynomial and rational matrices and their eigenstructures and continues by providing new scientific results in the form of four attached articles. In the first article, we derive new results for the conditioning of zeros of rational matrices. This is achieved by using the transfer function representation for rational matrices together with Rosenbrock's theorem, which relates the eigenstructure of a rational matrix with that of the associated polynomial system matrix. These results have significance in control theory, where rational matrices appear as transfer function matrices in a natural way. The remaining three articles deal with nearness problems associated with the eigenstructure of polynomial matrices. These include the problem of finding the distance to singularity and the problem of finding the distance to stability. For these problems, we use a novel approach that relies on Riemannian optimization. This approach leads to computational algorithms that often outperform the competitors in terms of the running time as well as the quality of the output.","Polynomiset ja rationaaliset matriisit ovat tärkeitä algebrallisia objekteja, joilla on monia käytännön sovelluksia, erityisesti säätöteoriassa. Kuten klassisempien reaaliluku- tai kompleksilukukunnan matriisien tapauksessa, myös polynomisten ja rationaalisten matriisien ominaisuuksia voidaan kuvata niiden ominaisrakenteen avulla. Tämä väitöskirja käsittelee polynomisten ja rationaalisten matriisien ominaisrakenteeseen liittyviä kysymyksiä. Erityisesti tutkimme, kuinka muutokset matriisien alkioissa vaikuttavat ominaisrakenteeseen ja päinvastoin. Nämä kysymykset kuuluvat yleensä joko häiriöteorian tai matriisien lähisyysongelmien piiriin ja ovat tärkeitä esimerkiksi arvioitaessa, kuinka paljon virhettä ominaisrakenteen numeerisessa laskennassa voi esiintyä. Väitöskirja alkaa antamalla yleiskatsauksen polynomisten ja rationaalisten matriisien sekä niiden ominaisrakenteiden teoriasta. Sen jälkeen esitellään uusia tieteellisiä tuloksia neljän liitteenä olevan artikkelin muodossa. Ensimmäisessä artikkelissa johdamme uusia tuloksia rationaalisten matriisien nollakohtien häiriöalttiudesta. Tämä saavutetaan hyödyntämällä rationaalisten matriisien siirtofunktioesitystä yhdessä Rosenbrockin teoreeman kanssa, joka yhdistää rationaalisen matriisin ominaisrakenteen siihen liittyvän polynomisen systeemimatriisin ominaisrakenteeseen. Näillä tuloksilla on merkitystä säätöteoriassa, jossa rationaaliset matriisit esiintyvät luonnollisella tavalla siirtofunktiomatriiseina. Muut kolme artikkelia käsittelevät polynomisten matriisien ominaisrakenteeseen liittyviä lähisyysongelmia. Näihin kuuluvat singulaarisuuteen ja stabilisuuteen liittyvien etäisyyksien määrittämisen ongelmat. Näitä varten käytämme uutta lähestymistapaa, joka perustuu Riemannin optimointiin. Tämä lähestymistapa johtaa laskennallisiin algoritmeihin, jotka usein päihittävät kilpailijat sekä laskenta-ajan että tulosten laadun osalta."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Eigenstructure of polynomial and rational matrices: perturbation theory and nearness problems","Polynomi- ja rationaalimatriisien ominaisrakenne: perturbaatioteoria ja läheisyysongelmat"]}]}],"canonical_facts":{"dc:contributor":["Aalto-yliopisto","Aalto University"],"dc:contributor.advisor":["Noferini, Vanni, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland"],"dc:contributor.department":["Matematiikan ja systeemianalyysin laitos","Department of Mathematics and Systems Analysis"],"dc:contributor.supervisor":["Noferini, Vanni, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland"],"dc:creator":["Nyman, Lauri"],"dc:date.accessioned":["2025-05-20T09:00:35Z"],"dc:date.available":["2025-05-20T09:00:35Z"],"dc:date.issued":["2025"],"dc:description.abstract":["Polynomial and rational matrices are important algebraic objects that have many engineering applications, most notably in control theory. As with more classical matrices over the field of real or complex numbers, many of the properties of polynomial and rational matrices can be described in terms of their eigenstructure. This thesis considers questions related to the eigenstructure of polynomial and rational matrices. In particular, we study how variations in the entries affect the eigenstructure, and vice versa. These questions generally fall under either perturbation theory or matrix nearness problems, and are important in, for example, understanding how much error there can be in a numerical computation of the eigenstructure. This thesis begins with an overview on the theory of polynomial and rational matrices and their eigenstructures and continues by providing new scientific results in the form of four attached articles. In the first article, we derive new results for the conditioning of zeros of rational matrices. This is achieved by using the transfer function representation for rational matrices together with Rosenbrock's theorem, which relates the eigenstructure of a rational matrix with that of the associated polynomial system matrix. These results have significance in control theory, where rational matrices appear as transfer function matrices in a natural way. The remaining three articles deal with nearness problems associated with the eigenstructure of polynomial matrices. These include the problem of finding the distance to singularity and the problem of finding the distance to stability. For these problems, we use a novel approach that relies on Riemannian optimization. This approach leads to computational algorithms that often outperform the competitors in terms of the running time as well as the quality of the output.","Polynomiset ja rationaaliset matriisit ovat tärkeitä algebrallisia objekteja, joilla on monia käytännön sovelluksia, erityisesti säätöteoriassa. Kuten klassisempien reaaliluku- tai kompleksilukukunnan matriisien tapauksessa, myös polynomisten ja rationaalisten matriisien ominaisuuksia voidaan kuvata niiden ominaisrakenteen avulla. Tämä väitöskirja käsittelee polynomisten ja rationaalisten matriisien ominaisrakenteeseen liittyviä kysymyksiä. Erityisesti tutkimme, kuinka muutokset matriisien alkioissa vaikuttavat ominaisrakenteeseen ja päinvastoin. Nämä kysymykset kuuluvat yleensä joko häiriöteorian tai matriisien lähisyysongelmien piiriin ja ovat tärkeitä esimerkiksi arvioitaessa, kuinka paljon virhettä ominaisrakenteen numeerisessa laskennassa voi esiintyä. Väitöskirja alkaa antamalla yleiskatsauksen polynomisten ja rationaalisten matriisien sekä niiden ominaisrakenteiden teoriasta. Sen jälkeen esitellään uusia tieteellisiä tuloksia neljän liitteenä olevan artikkelin muodossa. Ensimmäisessä artikkelissa johdamme uusia tuloksia rationaalisten matriisien nollakohtien häiriöalttiudesta. Tämä saavutetaan hyödyntämällä rationaalisten matriisien siirtofunktioesitystä yhdessä Rosenbrockin teoreeman kanssa, joka yhdistää rationaalisen matriisin ominaisrakenteen siihen liittyvän polynomisen systeemimatriisin ominaisrakenteeseen. Näillä tuloksilla on merkitystä säätöteoriassa, jossa rationaaliset matriisit esiintyvät luonnollisella tavalla siirtofunktiomatriiseina. Muut kolme artikkelia käsittelevät polynomisten matriisien ominaisrakenteeseen liittyviä lähisyysongelmia. Näihin kuuluvat singulaarisuuteen ja stabilisuuteen liittyvien etäisyyksien määrittämisen ongelmat. Näitä varten käytämme uutta lähestymistapaa, joka perustuu Riemannin optimointiin. Tämä lähestymistapa johtaa laskennallisiin algoritmeihin, jotka usein päihittävät kilpailijat sekä laskenta-ajan että tulosten laadun osalta."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://aaltodoc.aalto.fi/handle/123456789/135593"],"dc:language.iso":["en"],"dc:publisher":["Aalto University","Aalto-yliopisto"],"dc:title":["Eigenstructure of polynomial and rational matrices: perturbation theory and nearness problems","Polynomi- ja rationaalimatriisien ominaisrakenne: perturbaatioteoria ja läheisyysongelmat"],"dc:type":["G5 Artikkeliväitöskirja"],"dc:type.dcmitype":["text"]},"updated_at":"2026-08-21T22:21:56Z"}