Completing symplectic matrices

Let F be a field of characteristic not equal 2, V a non-singular 2n-dimensi onal symplectic space over F, nu(1), nu(2),..., nu(2n) a basis for V, and S p(n)(F) the collection of symplectic isometries of V with respect to this b asis. We consider the following completion question: If A is any n x n F-ma trix, must there be some D is an element of Sp(2n) (F) with D = (A * ) * *) It is shown that for some particular important choices of bases, the answer is yes, but it does not hold in general. (C) 2000 Elsevier Science Inc. Al l rights reserved.