On quaternionic numerical ranges with respect to nonstandard involutions
Let ϕ be a nonstandard involution on the set of all quaternions, and the quaternion α be such that ϕ(α)=α. The notion of numerical range of an n×n quaternion matrix A with respect to ϕ was introduced by Leiba Rodman (2014) [8] asWϕ(α)(A)={xϕAx:x is an n×1 quaternion vector andxϕx=α}, where for x=[x1...
Gespeichert in:
| Veröffentlicht in: | Linear algebra and its applications 2018-03, Vol.540, p.11-25 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 25 |
|---|---|
| container_issue | |
| container_start_page | 11 |
| container_title | Linear algebra and its applications |
| container_volume | 540 |
| creator | Aghamollaei, Gholamreza Rahjoo, Meysam |
| description | Let ϕ be a nonstandard involution on the set of all quaternions, and the quaternion α be such that ϕ(α)=α. The notion of numerical range of an n×n quaternion matrix A with respect to ϕ was introduced by Leiba Rodman (2014) [8] asWϕ(α)(A)={xϕAx:x is an n×1 quaternion vector andxϕx=α}, where for x=[x1⋯xn]T, xϕ=[ϕ(x1)⋯ϕ(xn)]. In this paper, some algebraic and geometrical properties of Wϕ(0)(.) for every arbitrary quaternion matrix are investigated. Moreover, a description of this set is given for 2×2 quaternion matrices, and Wϕ(0)(.) is characterized for ϕ-hermitian and ϕ-skewhermitian quaternion matrices. To illustrate the main results, some examples are also given. |
| doi_str_mv | 10.1016/j.laa.2017.11.013 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2082037662</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0024379517306419</els_id><sourcerecordid>2082037662</sourcerecordid><originalsourceid>FETCH-LOGICAL-c368t-1bb85c4c81d06f4414cddcd0e50c7c4f1ededd4fda37c51e546e9db532f7399c3</originalsourceid><addsrcrecordid>eNp9kD1PwzAQhi0EEqXwA9giMSf4YidOxYQqvqRKXWC2XPsCjlK7tZ0i_j2uysx0y_u8d_cQcgu0Agrt_VCNSlU1BVEBVBTYGZlBJ1gJXdOekxmlNS-ZWDSX5CrGgVLKBa1n5HXtiv2kEgZnvbO6cNMWg9VqLIJynxiLb5u-ioBxhzoVyRfOu5iUMyqYwrqDH6eUyXhNLno1Rrz5m3Py8fz0vnwtV-uXt-XjqtSs7VIJm03XaK47MLTtOQeujdGGYkO10LwHNGgM741iQjeADW9xYTYNq3vBFgvN5uTu1LsLfj9hTHLwU3B5paxpV1Mm2rbOKTildPAxBuzlLtitCj8SqDwKk4PMwuRRmASQWVhmHk4M5vMPFoOM2qLTaGzIv0vj7T_0L67pdMM</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2082037662</pqid></control><display><type>article</type><title>On quaternionic numerical ranges with respect to nonstandard involutions</title><source>Elsevier ScienceDirect Journals Complete</source><source>Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals</source><creator>Aghamollaei, Gholamreza ; Rahjoo, Meysam</creator><creatorcontrib>Aghamollaei, Gholamreza ; Rahjoo, Meysam</creatorcontrib><description>Let ϕ be a nonstandard involution on the set of all quaternions, and the quaternion α be such that ϕ(α)=α. The notion of numerical range of an n×n quaternion matrix A with respect to ϕ was introduced by Leiba Rodman (2014) [8] asWϕ(α)(A)={xϕAx:x is an n×1 quaternion vector andxϕx=α}, where for x=[x1⋯xn]T, xϕ=[ϕ(x1)⋯ϕ(xn)]. In this paper, some algebraic and geometrical properties of Wϕ(0)(.) for every arbitrary quaternion matrix are investigated. Moreover, a description of this set is given for 2×2 quaternion matrices, and Wϕ(0)(.) is characterized for ϕ-hermitian and ϕ-skewhermitian quaternion matrices. To illustrate the main results, some examples are also given.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2017.11.013</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Computer graphics ; Linear algebra ; Matrix ; Nonstandard involution ; Numerical range ; Quantum physics ; Quaternion matrices ; Quaternions ; Set theory</subject><ispartof>Linear algebra and its applications, 2018-03, Vol.540, p.11-25</ispartof><rights>2017 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. Mar 1, 2018</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c368t-1bb85c4c81d06f4414cddcd0e50c7c4f1ededd4fda37c51e546e9db532f7399c3</citedby><cites>FETCH-LOGICAL-c368t-1bb85c4c81d06f4414cddcd0e50c7c4f1ededd4fda37c51e546e9db532f7399c3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.laa.2017.11.013$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Aghamollaei, Gholamreza</creatorcontrib><creatorcontrib>Rahjoo, Meysam</creatorcontrib><title>On quaternionic numerical ranges with respect to nonstandard involutions</title><title>Linear algebra and its applications</title><description>Let ϕ be a nonstandard involution on the set of all quaternions, and the quaternion α be such that ϕ(α)=α. The notion of numerical range of an n×n quaternion matrix A with respect to ϕ was introduced by Leiba Rodman (2014) [8] asWϕ(α)(A)={xϕAx:x is an n×1 quaternion vector andxϕx=α}, where for x=[x1⋯xn]T, xϕ=[ϕ(x1)⋯ϕ(xn)]. In this paper, some algebraic and geometrical properties of Wϕ(0)(.) for every arbitrary quaternion matrix are investigated. Moreover, a description of this set is given for 2×2 quaternion matrices, and Wϕ(0)(.) is characterized for ϕ-hermitian and ϕ-skewhermitian quaternion matrices. To illustrate the main results, some examples are also given.</description><subject>Computer graphics</subject><subject>Linear algebra</subject><subject>Matrix</subject><subject>Nonstandard involution</subject><subject>Numerical range</subject><subject>Quantum physics</subject><subject>Quaternion matrices</subject><subject>Quaternions</subject><subject>Set theory</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp9kD1PwzAQhi0EEqXwA9giMSf4YidOxYQqvqRKXWC2XPsCjlK7tZ0i_j2uysx0y_u8d_cQcgu0Agrt_VCNSlU1BVEBVBTYGZlBJ1gJXdOekxmlNS-ZWDSX5CrGgVLKBa1n5HXtiv2kEgZnvbO6cNMWg9VqLIJynxiLb5u-ioBxhzoVyRfOu5iUMyqYwrqDH6eUyXhNLno1Rrz5m3Py8fz0vnwtV-uXt-XjqtSs7VIJm03XaK47MLTtOQeujdGGYkO10LwHNGgM741iQjeADW9xYTYNq3vBFgvN5uTu1LsLfj9hTHLwU3B5paxpV1Mm2rbOKTildPAxBuzlLtitCj8SqDwKk4PMwuRRmASQWVhmHk4M5vMPFoOM2qLTaGzIv0vj7T_0L67pdMM</recordid><startdate>20180301</startdate><enddate>20180301</enddate><creator>Aghamollaei, Gholamreza</creator><creator>Rahjoo, Meysam</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></search><sort><creationdate>20180301</creationdate><title>On quaternionic numerical ranges with respect to nonstandard involutions</title><author>Aghamollaei, Gholamreza ; Rahjoo, Meysam</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-1bb85c4c81d06f4414cddcd0e50c7c4f1ededd4fda37c51e546e9db532f7399c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Computer graphics</topic><topic>Linear algebra</topic><topic>Matrix</topic><topic>Nonstandard involution</topic><topic>Numerical range</topic><topic>Quantum physics</topic><topic>Quaternion matrices</topic><topic>Quaternions</topic><topic>Set theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Aghamollaei, Gholamreza</creatorcontrib><creatorcontrib>Rahjoo, Meysam</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>Aghamollaei, Gholamreza</au><au>Rahjoo, Meysam</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On quaternionic numerical ranges with respect to nonstandard involutions</atitle><jtitle>Linear algebra and its applications</jtitle><date>2018-03-01</date><risdate>2018</risdate><volume>540</volume><spage>11</spage><epage>25</epage><pages>11-25</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>Let ϕ be a nonstandard involution on the set of all quaternions, and the quaternion α be such that ϕ(α)=α. The notion of numerical range of an n×n quaternion matrix A with respect to ϕ was introduced by Leiba Rodman (2014) [8] asWϕ(α)(A)={xϕAx:x is an n×1 quaternion vector andxϕx=α}, where for x=[x1⋯xn]T, xϕ=[ϕ(x1)⋯ϕ(xn)]. In this paper, some algebraic and geometrical properties of Wϕ(0)(.) for every arbitrary quaternion matrix are investigated. Moreover, a description of this set is given for 2×2 quaternion matrices, and Wϕ(0)(.) is characterized for ϕ-hermitian and ϕ-skewhermitian quaternion matrices. To illustrate the main results, some examples are also given.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2017.11.013</doi><tpages>15</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0024-3795 |
| ispartof | Linear algebra and its applications, 2018-03, Vol.540, p.11-25 |
| issn | 0024-3795 1873-1856 |
| language | eng |
| recordid | cdi_proquest_journals_2082037662 |
| source | Elsevier ScienceDirect Journals Complete; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Computer graphics Linear algebra Matrix Nonstandard involution Numerical range Quantum physics Quaternion matrices Quaternions Set theory |
| title | On quaternionic numerical ranges with respect to nonstandard involutions |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-04T12%3A18%3A58IST&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=On%20quaternionic%20numerical%20ranges%20with%20respect%20to%20nonstandard%20involutions&rft.jtitle=Linear%20algebra%20and%20its%20applications&rft.au=Aghamollaei,%20Gholamreza&rft.date=2018-03-01&rft.volume=540&rft.spage=11&rft.epage=25&rft.pages=11-25&rft.issn=0024-3795&rft.eissn=1873-1856&rft_id=info:doi/10.1016/j.laa.2017.11.013&rft_dat=%3Cproquest_cross%3E2082037662%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=2082037662&rft_id=info:pmid/&rft_els_id=S0024379517306419&rfr_iscdi=true |