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In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is * a ''bilinear form'': a map that is bilinear (i.e. linear in each argument separately), * ''alternating'': holds for all , and * ''nondegenerate'': for all implies that is zero. If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above say that this matrix must be skew-symmetric, nonsingular, and hollow. 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, hollow matrix of odd size has determinant zero. Notice the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces. ==Standard symplectic space== The standard symplectic space is R2''n'' with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically ''ω'' is chosen to be the block matrix : where ''I''''n'' is the identity matrix. In terms of basis vectors : : : A modified version of the Gram–Schmidt process shows that any finite-dimensional symplectic vector space has a basis such that ''ω'' takes this form, often called a ''Darboux basis'', or symplectic basis. There is another way to interpret this standard symplectic form. Since the model space R''n'' used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let ''V'' be a real vector space of dimension ''n'' and ''V''∗ its dual space. Now consider the direct sum of these spaces equipped with the following form: : Now choose any basis of ''V'' and consider its dual basis : We can interpret the basis vectors as lying in ''W'' if we write . Taken together, these form a complete basis of ''W'', : The form ''ω'' defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form . The subspace ''V'' is not unique, and a choice of subspace ''V'' is called a polarization. The subspaces that give such an isomorphism are called Lagrangian subspaces or simply Lagrangians. Explicitly, given a Lagrangian subspace (as defined below), then a choice of basis defines a dual basis for a complement, by . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「symplectic vector space」の詳細全文を読む スポンサード リンク
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