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

AbstractWe provide a necessary and sufficient 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 find 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 sufficiency 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 [0−111]. The provided condition improves the one in [Borobia A, Canogar R, De Terán F. 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⩽min{n−nA−rank(A+AT)2,rank(A−AT)}, where nA is the dimension of Nul A∩Nul AT.Keywords: Matrix equationconsistencytransposecongruencecanonical form for congruenceskew-symmetric matrixsymplectic bilinear formAMS Subject Classifications: 15A2115A2415A63 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work has been funded by the Agencia Estatal de Investigación 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 Terán).