2n-by-2n symplectic completions of matrices of order 2n - 1

Let F be a field. Set J := [GRAPHICS] . Amatrix X is an element of F-2nx2n is symplectic if XJ X-1(T) J = I. We say that a matrix P is an element of F2n-1x2n-1 has a symplectic completion of order 2n if there exist x, y is an element of F2n-1x1 and a scalar alpha is an element of F such that [GRAPHICS] is symplectic. If P is nonsingular, we give necessary and sufficient conditions such that P has a symplectic completion of order 2n. We give an implicit characterization of all matrices of order 3 which have symplectic completions of order 4.