Complementary bases in symplectic matrices and a proof that their determinant is one
It is well known that symplectic transformations have determinant \(+1\). E.g., this follows immediately from the fact that the symplectic group is generated by symplectic transvections, but various other proofs are known. The authors give a very short direct proof for the fact the determinant of a symplectic matrix is \(+1\) under the extra assumption that the given matrix has a certain block form with one invertible block. Next, linearly independent rows and columns of a symplectic matrix are exhibited which then allows to reduce the case of an arbitrary symplectic matrix to the particular case mentioned in the above.
scientific article; zbMATH DE number 5126044