Completing symplectic matrices (Q1573660)

Completing symplectic matrices \(D\in Sp_{2n}\) is studied \((Sp_{2n}\) is the symplectic group, \(F\) is a field of characteristic \(\neq 2\), \(V\) a non-singular \(2n\)-dimensional symplectic space over \(F\)). Choices of bases of \(V\) are discussed for the existence of \(D=|\begin{smallmatrix} A & *\\ * & *\end{smallmatrix} |\), \(D\in Sp_{2n}(F)\). It holds for \(A\) if \(r\leq n\), generally it does not hold.