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On the consistency of the matrix equation XTAX B when B is skew-symmetric: improving the previous characterization

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dc.contributor.authorBorobia Vizmanos, Alberto
dc.contributor.authorCanogar Mckenzie, Roberto
dc.contributor.authorTeran, Fernando De
dc.contributor.funderMinisterio de Ciencia e Innovación de España
dc.date.accessioned2025-12-05T15:56:32Z
dc.date.available2025-12-05T15:56:32Z
dc.date.issued2023-05-16
dc.descriptionThe registered version of this article, first published in Linear and Multilinear Algebra, is available online at the publisher's website: Taylor and Francis Group, https://doi.org/10.1080/03081087.2023.2211720
dc.descriptionLa versión registrada de este artículo, publicado por primera vez en Linear and Multilinear Algebra, está disponible en línea en el sitio web del editor: Taylor and Francis Group, https://doi.org/10.1080/03081087.2023.2211720
dc.description.abstractWe provide a necessary and su cient condition for the matrix equation XTAX B to be consistent, when A is an arbitrary complex square matrix and B is skew-symmetric. This problem is equivalent to nd the largest dimension of a subspace in which the bilinear form A is symplectic. The necessity is valid for any A and B as above, whereas the su ciency is proved to be valid for any skew-symmetric matrix B and for all complex square matrices A whose Canonical form for congruence (CFC) does not contain blocks 0 1 1 1 . The provided condition improves the one in [A. Borobia, R. Canogar, F. De Teran, Lin. Multilin. Algebra, 2022. doi:10.1080/03081087.2022.2093825], because it includes the case where CFC(A) includes symmetric blocks, and it is given in terms of the size of A and the rank of its symmetric and skew-symmetric parts. More precisely, if A is n n, we prove that the equation is consistent if and only if rankB mintn nA rankpA ATq 2 NulAXNulAT.en
dc.description.sponsorshipThis work has been funded by the Agencia Estatal de Investigacion of Spain through grant PID2019-106362GB-I00/AEI/10.13039/501100011033 and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation) (Fernando De Teran).
dc.description.versionversión final
dc.identifier.citationBorobia, A., Canogar, R., & De Terán, F. (2025). On the consistency of the matrix equation XTA X = B when B is skew-symmetric: improving the previous characterization. Linear and Multilinear Algebra, 73(9), 1820–1846. https://doi.org/10.1080/03081087.2023.2211720
dc.identifier.doihttps://doi.org/10.1080/03081087.2023.2211720
dc.identifier.issn1563-5139
dc.identifier.urihttps://hdl.handle.net/20.500.14468/31036
dc.journal.issue9
dc.journal.titleLinear and Multilinear Algebra
dc.journal.volume73
dc.language.isoen
dc.page.final1846
dc.page.initial1820
dc.publisherTaylor and Francis Group
dc.relation.centerFacultad de Ciencias
dc.relation.departmentMatemáticas Fundamentales
dc.relation.projectidinfo:eu-repo/grantAgreement/MICINN/Programa Estatal de Investigación, Desarrollo e Innovación Orientada a los Retos de la Sociedad/PID2019-106362GB-I00
dc.rightsinfo:eu-repo/semantics/openAccess
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
dc.subject12 Matemáticas
dc.subject.keywordsMatrix equationen
dc.subject.keywordsconsistencyen
dc.subject.keywordstransposeen
dc.subject.keywordscongruenceen
dc.subject.keywordsCanonical form for congruenceen
dc.subject.keywordsskew-symmetric matrixen
dc.subject.keywordssymplectic bilinear formen
dc.titleOn the consistency of the matrix equation XTAX B when B is skew-symmetric: improving the previous characterizationen
dc.typeartículoes
dc.typejournal articleen
dspace.entity.typePublication
relation.isAuthorOfPublicationfda35eed-831c-4588-9504-f202a8c957f6
relation.isAuthorOfPublication6222b7d3-ae1d-411e-914a-a59610f0a254
relation.isAuthorOfPublication.latestForDiscoveryfda35eed-831c-4588-9504-f202a8c957f6
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