|
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector space ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector for that quadratic form. Suppose that is quadratic space and ''W'' is a subspace. Then ''W'' is called an isotropic subspace of ''V'' if ''some'' vector in it is isotropic, a totally isotropic subspace if ''all'' vectors in it are isotropic, and an anisotropic subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.〔 A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a definite form: : * either ''q'' is ''positive definite'', i.e. for all non-zero ''v'' in ''V'' ; : * or ''q'' is ''negative definite'', i.e. for all non-zero ''v'' in ''V''. More generally, if the quadratic form is non-degenerate and has the signature , then its isotropy index is the minimum of ''a'' and ''b''. ==Hyperbolic plane== Let with elements . Then the quadratic forms and are equivalent since there is a linear transformation on ''V'' that makes ''q'' look like ''r'', and vice versa. Evidently, and are isotropic. This example is called the hyperbolic plane in the theory of quadratic forms. A common instance has ''F'' = real numbers in which case and are hyperbolas. In particular, is the unit hyperbola. The notation has been used by Milnor and Huseman〔Milnor & Husemoller (1973) page 9〕 for the hyperbolic plane as the signs of the terms of the bivariate polynomial ''r'' are exhibited. 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「Isotropic quadratic form」の詳細全文を読む スポンサード リンク
|