Geometric mean block matrices
We consider an m×m block matrix G with entries Ai#Aj where A1,…,Am are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A1,…,Am is Γ-commuting; it can be transformed to a c...
Gespeichert in:
| Veröffentlicht in: | Linear algebra and its applications 2019-08, Vol.575, p.299-313 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 313 |
|---|---|
| container_issue | |
| container_start_page | 299 |
| container_title | Linear algebra and its applications |
| container_volume | 575 |
| creator | Kim, Sejong Lee, Hosoo Lim, Yongdo |
| description | We consider an m×m block matrix G with entries Ai#Aj where A1,…,Am are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A1,…,Am is Γ-commuting; it can be transformed to a commuting family of positive definite matrices by a congruence transformation. This result via Γ-commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices. |
| doi_str_mv | 10.1016/j.laa.2019.04.008 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2246255227</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0024379519301600</els_id><sourcerecordid>2246255227</sourcerecordid><originalsourceid>FETCH-LOGICAL-c368t-a1f47e284cabe635b4bce7f45e14f52543cb634c9de70e7ca3f8d55a2ce37e3c3</originalsourceid><addsrcrecordid>eNp9kMtKxEAQRRtRMI5-gAsh4Dqx-pVOcCWDMwoDbnTddCoVSMxj7M4I_r0Jce2qoLjnVnEYu-WQcuDZQ5t2zqUCeJGCSgHyMxbx3MiE5zo7ZxGAUIk0hb5kVyG0AKAMiIjd7WnsafINxj25IS67ET_j3i0bCtfsonZdoJu_uWEfu-f37UtyeNu_bp8OCcosnxLHa2VI5ApdSZnUpSqRTK00cVVroZXEMpMKi4oMkEEn67zS2gkkaUii3LD7tffox68Thcm248kP80krhMqE1kKYOcXXFPoxBE-1Pfqmd_7HcrCLBdva2YJdLFhQdrYwM48rQ_P73w15G7ChAalqPOFkq7H5h_4F9N9jeg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2246255227</pqid></control><display><type>article</type><title>Geometric mean block matrices</title><source>Elsevier ScienceDirect Journals Complete</source><source>EZB-FREE-00999 freely available EZB journals</source><creator>Kim, Sejong ; Lee, Hosoo ; Lim, Yongdo</creator><creatorcontrib>Kim, Sejong ; Lee, Hosoo ; Lim, Yongdo</creatorcontrib><description>We consider an m×m block matrix G with entries Ai#Aj where A1,…,Am are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A1,…,Am is Γ-commuting; it can be transformed to a commuting family of positive definite matrices by a congruence transformation. This result via Γ-commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2019.04.008</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Ando-Li-Mathias geometric mean ; Block matrix ; Commuting ; Geometric mean ; Linear algebra ; Mathematical analysis ; Matrix methods ; Positive definite matrix ; Γ-commuting family</subject><ispartof>Linear algebra and its applications, 2019-08, Vol.575, p.299-313</ispartof><rights>2019 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. Aug 15, 2019</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c368t-a1f47e284cabe635b4bce7f45e14f52543cb634c9de70e7ca3f8d55a2ce37e3c3</citedby><cites>FETCH-LOGICAL-c368t-a1f47e284cabe635b4bce7f45e14f52543cb634c9de70e7ca3f8d55a2ce37e3c3</cites><orcidid>0000-0002-4576-734X ; 0000-0002-5181-4587</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.laa.2019.04.008$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Kim, Sejong</creatorcontrib><creatorcontrib>Lee, Hosoo</creatorcontrib><creatorcontrib>Lim, Yongdo</creatorcontrib><title>Geometric mean block matrices</title><title>Linear algebra and its applications</title><description>We consider an m×m block matrix G with entries Ai#Aj where A1,…,Am are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A1,…,Am is Γ-commuting; it can be transformed to a commuting family of positive definite matrices by a congruence transformation. This result via Γ-commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices.</description><subject>Ando-Li-Mathias geometric mean</subject><subject>Block matrix</subject><subject>Commuting</subject><subject>Geometric mean</subject><subject>Linear algebra</subject><subject>Mathematical analysis</subject><subject>Matrix methods</subject><subject>Positive definite matrix</subject><subject>Γ-commuting family</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKxEAQRRtRMI5-gAsh4Dqx-pVOcCWDMwoDbnTddCoVSMxj7M4I_r0Jce2qoLjnVnEYu-WQcuDZQ5t2zqUCeJGCSgHyMxbx3MiE5zo7ZxGAUIk0hb5kVyG0AKAMiIjd7WnsafINxj25IS67ET_j3i0bCtfsonZdoJu_uWEfu-f37UtyeNu_bp8OCcosnxLHa2VI5ApdSZnUpSqRTK00cVVroZXEMpMKi4oMkEEn67zS2gkkaUii3LD7tffox68Thcm248kP80krhMqE1kKYOcXXFPoxBE-1Pfqmd_7HcrCLBdva2YJdLFhQdrYwM48rQ_P73w15G7ChAalqPOFkq7H5h_4F9N9jeg</recordid><startdate>20190815</startdate><enddate>20190815</enddate><creator>Kim, Sejong</creator><creator>Lee, Hosoo</creator><creator>Lim, Yongdo</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><orcidid>https://orcid.org/0000-0002-4576-734X</orcidid><orcidid>https://orcid.org/0000-0002-5181-4587</orcidid></search><sort><creationdate>20190815</creationdate><title>Geometric mean block matrices</title><author>Kim, Sejong ; Lee, Hosoo ; Lim, Yongdo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-a1f47e284cabe635b4bce7f45e14f52543cb634c9de70e7ca3f8d55a2ce37e3c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Ando-Li-Mathias geometric mean</topic><topic>Block matrix</topic><topic>Commuting</topic><topic>Geometric mean</topic><topic>Linear algebra</topic><topic>Mathematical analysis</topic><topic>Matrix methods</topic><topic>Positive definite matrix</topic><topic>Γ-commuting family</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kim, Sejong</creatorcontrib><creatorcontrib>Lee, Hosoo</creatorcontrib><creatorcontrib>Lim, Yongdo</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>Kim, Sejong</au><au>Lee, Hosoo</au><au>Lim, Yongdo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Geometric mean block matrices</atitle><jtitle>Linear algebra and its applications</jtitle><date>2019-08-15</date><risdate>2019</risdate><volume>575</volume><spage>299</spage><epage>313</epage><pages>299-313</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>We consider an m×m block matrix G with entries Ai#Aj where A1,…,Am are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A1,…,Am is Γ-commuting; it can be transformed to a commuting family of positive definite matrices by a congruence transformation. This result via Γ-commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2019.04.008</doi><tpages>15</tpages><orcidid>https://orcid.org/0000-0002-4576-734X</orcidid><orcidid>https://orcid.org/0000-0002-5181-4587</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0024-3795 |
| ispartof | Linear algebra and its applications, 2019-08, Vol.575, p.299-313 |
| issn | 0024-3795 1873-1856 |
| language | eng |
| recordid | cdi_proquest_journals_2246255227 |
| source | Elsevier ScienceDirect Journals Complete; EZB-FREE-00999 freely available EZB journals |
| subjects | Ando-Li-Mathias geometric mean Block matrix Commuting Geometric mean Linear algebra Mathematical analysis Matrix methods Positive definite matrix Γ-commuting family |
| title | Geometric mean block matrices |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-07T20%3A51%3A40IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Geometric%20mean%20block%20matrices&rft.jtitle=Linear%20algebra%20and%20its%20applications&rft.au=Kim,%20Sejong&rft.date=2019-08-15&rft.volume=575&rft.spage=299&rft.epage=313&rft.pages=299-313&rft.issn=0024-3795&rft.eissn=1873-1856&rft_id=info:doi/10.1016/j.laa.2019.04.008&rft_dat=%3Cproquest_cross%3E2246255227%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2246255227&rft_id=info:pmid/&rft_els_id=S0024379519301600&rfr_iscdi=true |