Borobia Vizmanos, AlbertoCanogar Mckenzie, RobertoTeran, Fernando De2025-12-052025-12-052023-05-16Borobia, 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/10.1080/03081087.2023.22117201563-5139https://doi.org/10.1080/03081087.2023.2211720https://hdl.handle.net/20.500.14468/31036The 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/10.1080/03081087.2023.2211720La 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/10.1080/03081087.2023.2211720We 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.eninfo:eu-repo/semantics/openAccess12 MatemáticasartículoMatrix equationconsistencytransposecongruenceCanonical form for congruenceskew-symmetric matrixsymplectic bilinear form