Symplectic basis

In linear algebra, a standard symplectic basis is a basis ${{\mathbf e}}_{i},{{\mathbf f}}_{i}$ of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form $\omega$ , such that $\omega ({{\mathbf e}}_{i},{{\mathbf e}}_{j})=0=\omega ({{\mathbf f}}_{i},{{\mathbf f}}_{j}),\omega ({{\mathbf e}}_{i},{{\mathbf f}}_{j})=\delta _{{ij}}$ . A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the Gram–Schmidt process.^[1] The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.

Notes

↑ Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006), p.7 and pp. 12–13

References

da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). ISBN 3-540-42195-5.
Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 978-3-7643-7574-4.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Symplectic basis

See also

Notes

References