Abstract
We provide a necessary and sufficient condition for the matrix equation 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 . 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 , we prove that the equation is consistent if and only if , where is the dimension of .
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