Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form called the symplectic form.Working in a fixed basis, can be represented by a matrix. The two conditions above say that this matrix must be skew-symmetric and nonsingular. This is not the same thing as a symplectic matrix, which represents a symplectic transformation of the space.If V is finite-dimensional then its dimension must necessarily be even since every skew-symmetric matrix of odd size has determinant zero. A nondegenerate skew-symmetric bilinear form behaves quite differently from a nondegenerate symmetric bilinear form, such as the dot product on Euclidean vector spaces. With a Euclidean inner product g, we have g(v,v) > 0 for all nonzero vectors v, whereas a symplectic form satisfies (v,v) = 0.

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