Exact Relation between Singular Value and Eigenvalue Statistics
We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one...
Gespeichert in:
| Veröffentlicht in: | arXiv.org 2016-02 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | |
|---|---|
| container_issue | |
| container_start_page | |
| container_title | arXiv.org |
| container_volume | |
| creator | Kieburg, Mario Kösters, Holger |
| description | We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance. |
| doi_str_mv | 10.48550/arxiv.1601.02586 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_arxiv</sourceid><recordid>TN_cdi_arxiv_primary_1601_02586</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2075299931</sourcerecordid><originalsourceid>FETCH-LOGICAL-a521-5dfc9957248cea2d9cc813ac43238c835e28ad90fb5f1a963e76920738367eda3</originalsourceid><addsrcrecordid>eNotj11LwzAYhYMgOOZ-gFcWvG5N3rdJkyuRUT9gILjhbXmXpqOjprNN5_z31s6rw4GHw3kYuxE8SbWU_J66U31MhOIi4SC1umAzQBSxTgGu2KLv95xzUBlIiTP2kJ_IhujdNRTq1kdbF76d89G69ruhoS76oGZwEfkyyuud88eprsNI96G2_TW7rKjp3eI_52zzlG-WL_Hq7fl1-biKSYKIZVlZY2QGqbaOoDTWaoFkUwTUVqN0oKk0vNrKSpBR6DJlgGeoUWWuJJyz2_PsZFccuvqTup_iz7KYLEfi7kwcuvZrcH0o9u3Q-fFTMQ5JMMagwF94aVQx</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2075299931</pqid></control><display><type>article</type><title>Exact Relation between Singular Value and Eigenvalue Statistics</title><source>arXiv.org</source><source>Free E- Journals</source><creator>Kieburg, Mario ; Kösters, Holger</creator><creatorcontrib>Kieburg, Mario ; Kösters, Holger</creatorcontrib><description>We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.1601.02586</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Deformation ; Eigenvalues ; Fourier analysis ; Functions (mathematics) ; Harmonic analysis ; Invariants ; Mathematical analysis ; Mathematics - Classical Analysis and ODEs ; Mathematics - Mathematical Physics ; Mathematics - Probability ; Physics - Mathematical Physics ; Polynomials</subject><ispartof>arXiv.org, 2016-02</ispartof><rights>2016. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,780,881,27902</link.rule.ids><backlink>$$Uhttps://doi.org/10.48550/arXiv.1601.02586$$DView paper in arXiv$$Hfree_for_read</backlink><backlink>$$Uhttps://doi.org/10.1142/S2010326316500155$$DView published paper (Access to full text may be restricted)$$Hfree_for_read</backlink></links><search><creatorcontrib>Kieburg, Mario</creatorcontrib><creatorcontrib>Kösters, Holger</creatorcontrib><title>Exact Relation between Singular Value and Eigenvalue Statistics</title><title>arXiv.org</title><description>We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.</description><subject>Deformation</subject><subject>Eigenvalues</subject><subject>Fourier analysis</subject><subject>Functions (mathematics)</subject><subject>Harmonic analysis</subject><subject>Invariants</subject><subject>Mathematical analysis</subject><subject>Mathematics - Classical Analysis and ODEs</subject><subject>Mathematics - Mathematical Physics</subject><subject>Mathematics - Probability</subject><subject>Physics - Mathematical Physics</subject><subject>Polynomials</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><sourceid>GOX</sourceid><recordid>eNotj11LwzAYhYMgOOZ-gFcWvG5N3rdJkyuRUT9gILjhbXmXpqOjprNN5_z31s6rw4GHw3kYuxE8SbWU_J66U31MhOIi4SC1umAzQBSxTgGu2KLv95xzUBlIiTP2kJ_IhujdNRTq1kdbF76d89G69ruhoS76oGZwEfkyyuud88eprsNI96G2_TW7rKjp3eI_52zzlG-WL_Hq7fl1-biKSYKIZVlZY2QGqbaOoDTWaoFkUwTUVqN0oKk0vNrKSpBR6DJlgGeoUWWuJJyz2_PsZFccuvqTup_iz7KYLEfi7kwcuvZrcH0o9u3Q-fFTMQ5JMMagwF94aVQx</recordid><startdate>20160205</startdate><enddate>20160205</enddate><creator>Kieburg, Mario</creator><creator>Kösters, Holger</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20160205</creationdate><title>Exact Relation between Singular Value and Eigenvalue Statistics</title><author>Kieburg, Mario ; Kösters, Holger</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a521-5dfc9957248cea2d9cc813ac43238c835e28ad90fb5f1a963e76920738367eda3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Deformation</topic><topic>Eigenvalues</topic><topic>Fourier analysis</topic><topic>Functions (mathematics)</topic><topic>Harmonic analysis</topic><topic>Invariants</topic><topic>Mathematical analysis</topic><topic>Mathematics - Classical Analysis and ODEs</topic><topic>Mathematics - Mathematical Physics</topic><topic>Mathematics - Probability</topic><topic>Physics - Mathematical Physics</topic><topic>Polynomials</topic><toplevel>online_resources</toplevel><creatorcontrib>Kieburg, Mario</creatorcontrib><creatorcontrib>Kösters, Holger</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kieburg, Mario</au><au>Kösters, Holger</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Exact Relation between Singular Value and Eigenvalue Statistics</atitle><jtitle>arXiv.org</jtitle><date>2016-02-05</date><risdate>2016</risdate><eissn>2331-8422</eissn><abstract>We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.1601.02586</doi><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | EISSN: 2331-8422 |
| ispartof | arXiv.org, 2016-02 |
| issn | 2331-8422 |
| language | eng |
| recordid | cdi_arxiv_primary_1601_02586 |
| source | arXiv.org; Free E- Journals |
| subjects | Deformation Eigenvalues Fourier analysis Functions (mathematics) Harmonic analysis Invariants Mathematical analysis Mathematics - Classical Analysis and ODEs Mathematics - Mathematical Physics Mathematics - Probability Physics - Mathematical Physics Polynomials |
| title | Exact Relation between Singular Value and Eigenvalue Statistics |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-04T20%3A19%3A59IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_arxiv&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Exact%20Relation%20between%20Singular%20Value%20and%20Eigenvalue%20Statistics&rft.jtitle=arXiv.org&rft.au=Kieburg,%20Mario&rft.date=2016-02-05&rft.eissn=2331-8422&rft_id=info:doi/10.48550/arxiv.1601.02586&rft_dat=%3Cproquest_arxiv%3E2075299931%3C/proquest_arxiv%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2075299931&rft_id=info:pmid/&rfr_iscdi=true |