An Elementary Proof That Symplectic Matrices Have Determinant One

Abstract

We give one more proof of the fact that symplectic matrices over real and complex fields have determinant one. While this has already been proved many times, there has been lasting interest in finding an elementary proof. Our result is restricted to the real and complex case due to its reliance on field-dependent spectral theory, however in this setting we obtain a proof which is more elementary in the sense that it is direct and requires only well-known facts. Finally, an explicit formula for the determinant of conjugate symplectic matrices in terms of its square subblocks is given.