Specht's criterion for systems of linear mappings
W. Specht (1940) proved that two n×n complex matrices A and B are unitarily similar if and only if tracew(A,A⁎)=tracew(B,B⁎) for every word w(x,y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consis...
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| Veröffentlicht in: | Linear algebra and its applications 2017-04, Vol.519, p.278-295 |
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| creator | Futorny, Vyacheslav Horn, Roger A. Sergeichuk, Vladimir V. |
| description | W. Specht (1940) proved that two n×n complex matrices A and B are unitarily similar if and only if tracew(A,A⁎)=tracew(B,B⁎) for every word w(x,y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph Q(A), whose vertices are inner product spaces and arrows are linear mappings. Denote by Q˜(A) the directed graph obtained by enlarging to Q(A) the adjoint linear mappings. We prove that a system A is transformed by isometries of its spaces to a system B if and only if the traces of all closed directed walks in Q˜(A) and Q˜(B) coincide. |
| doi_str_mv | 10.1016/j.laa.2017.01.006 |
| format | Article |
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Specht (1940) proved that two n×n complex matrices A and B are unitarily similar if and only if tracew(A,A⁎)=tracew(B,B⁎) for every word w(x,y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph Q(A), whose vertices are inner product spaces and arrows are linear mappings. Denote by Q˜(A) the directed graph obtained by enlarging to Q(A) the adjoint linear mappings. We prove that a system A is transformed by isometries of its spaces to a system B if and only if the traces of all closed directed walks in Q˜(A) and Q˜(B) coincide.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2017.01.006</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Criteria ; Graph theory ; Graphs ; Linear algebra ; Mathematical analysis ; Matrix ; Matrix methods ; Specht's criterion ; Unitary and Euclidean representations of quivers ; Unitary similarity ; Variables</subject><ispartof>Linear algebra and its applications, 2017-04, Vol.519, p.278-295</ispartof><rights>2017 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. 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Specht (1940) proved that two n×n complex matrices A and B are unitarily similar if and only if tracew(A,A⁎)=tracew(B,B⁎) for every word w(x,y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph Q(A), whose vertices are inner product spaces and arrows are linear mappings. Denote by Q˜(A) the directed graph obtained by enlarging to Q(A) the adjoint linear mappings. We prove that a system A is transformed by isometries of its spaces to a system B if and only if the traces of all closed directed walks in Q˜(A) and Q˜(B) coincide.</description><subject>Criteria</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Linear algebra</subject><subject>Mathematical analysis</subject><subject>Matrix</subject><subject>Matrix methods</subject><subject>Specht's criterion</subject><subject>Unitary and Euclidean representations of quivers</subject><subject>Unitary similarity</subject><subject>Variables</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp9kD1PwzAQhi0EEqXwA9giMTAlnO3GH2JCFV9SJQZgtlznDI7aONgpEv8eV2VmuuV93rt7CLmk0FCg4qZvNtY2DKhsgDYA4ojMqJK8pqoVx2QGwBY1l7o9JWc59wCwkMBmhL6O6D6n61y5FCZMIQ6Vj6nKP3nCba6irzZhQJuqrR3HMHzkc3Li7Sbjxd-ck_eH-7flU716eXxe3q1qx4Waas4EcxraDhWHjnkmhfZKaqE8COW80oKunXawaNdCSqk1oAOLHhT34Byfk6tD75ji1w7zZPq4S0NZaRgoLZksT5QUPaRcijkn9GZMYWvTj6Fg9mZMb4oZszdjgJpipjC3BwbL-d8Bk8ku4OCwCwndZLoY_qF_AbEbabg</recordid><startdate>20170415</startdate><enddate>20170415</enddate><creator>Futorny, Vyacheslav</creator><creator>Horn, Roger A.</creator><creator>Sergeichuk, Vladimir V.</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/0000-0002-1853-6618</orcidid></search><sort><creationdate>20170415</creationdate><title>Specht's criterion for systems of linear mappings</title><author>Futorny, Vyacheslav ; Horn, Roger A. ; Sergeichuk, Vladimir V.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-3262c905de830d2f2769f87968f068cf8961bc9c045b6777990ec0aef083f0cc3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Criteria</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Linear algebra</topic><topic>Mathematical analysis</topic><topic>Matrix</topic><topic>Matrix methods</topic><topic>Specht's criterion</topic><topic>Unitary and Euclidean representations of quivers</topic><topic>Unitary similarity</topic><topic>Variables</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Futorny, Vyacheslav</creatorcontrib><creatorcontrib>Horn, Roger A.</creatorcontrib><creatorcontrib>Sergeichuk, Vladimir V.</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>Futorny, Vyacheslav</au><au>Horn, Roger A.</au><au>Sergeichuk, Vladimir V.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Specht's criterion for systems of linear mappings</atitle><jtitle>Linear algebra and its applications</jtitle><date>2017-04-15</date><risdate>2017</risdate><volume>519</volume><spage>278</spage><epage>295</epage><pages>278-295</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>W. Specht (1940) proved that two n×n complex matrices A and B are unitarily similar if and only if tracew(A,A⁎)=tracew(B,B⁎) for every word w(x,y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph Q(A), whose vertices are inner product spaces and arrows are linear mappings. Denote by Q˜(A) the directed graph obtained by enlarging to Q(A) the adjoint linear mappings. We prove that a system A is transformed by isometries of its spaces to a system B if and only if the traces of all closed directed walks in Q˜(A) and Q˜(B) coincide.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2017.01.006</doi><tpages>18</tpages><orcidid>https://orcid.org/0000-0002-1853-6618</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Criteria Graph theory Graphs Linear algebra Mathematical analysis Matrix Matrix methods Specht's criterion Unitary and Euclidean representations of quivers Unitary similarity Variables |
| title | Specht's criterion for systems of linear mappings |
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