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Complementary bases in symplectic matrices and a proof that their determinant is one. (English) Zbl 1112.15002

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.

MSC:

15A03 Vector spaces, linear dependence, rank, lineability
15B57 Hermitian, skew-Hermitian, and related matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A09 Theory of matrix inversion and generalized inverses
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