symplectic basis

In linear algebra, a standard symplectic basis is a basis $\{\backslash mathbf\; e\}\_i,\; \{\backslash mathbf\; f\}\_i$ of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form $\backslash omega$, such that $\backslash omega(\{\backslash mathbf\; e\}\_i,\; \{\backslash mathbf\; e\}\_j)\; =\; 0\; =\; \backslash omega(\{\backslash mathbf\; f\}\_i,\; \{\backslash mathbf\; f\}\_j),\; \backslash omega(\{\backslash mathbf\; e\}\_i,\; \{\backslash mathbf\; f\}\_j)\; =\; \backslash delta\_\{ij\}$. A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the Gram–Schmidt process.Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006), p.7 and pp. 12–13 The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.