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| In mathematics, a [[quadratic form]] over a [[field (mathematics)|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 {{nowrap|1=''q''(''v'') = 0}}. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector for that quadratic form.
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| Suppose that (''V'',''q'') is [[quadratic space]] and ''W'' is a [[linear subspace|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 '''{{visible anchor|isotropy index}}''' of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.<ref name=MH57/>
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| A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a [[definite bilinear form|definite form]]:
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| :* either ''q'' is ''positive definite'', i.e. {{nowrap|1=''q''(''v'') > 0}} for all non-zero ''v'' in ''V'' ;
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| :* or ''q'' is ''negative definite'', i.e. {{nowrap|1=''q''(''v'') < 0}} for all non-zero ''v'' in ''V''.
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| More generally, if the quadratic form is non-degenerate and has the [[signature (quadratic form)|signature]] (''a'',''b''), then its isotropy index is the minimum of ''a'' and ''b''.
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| ==Hyperbolic plane==
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| Let ''V'' = ''F''<sup>2</sup> with elements (''x,y''). Then the quadratic forms ''q = xy'' and ''r'' = ''x''<sup>2</sup> − ''y''<sup>2</sup>
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| are equivalent since there is a [[linear transformation]] on ''V'' that makes ''q'' look like ''r'', and vice versa. Evidently (''V,q'') and (''V,r'') are isotropic. This example is called the '''hyperbolic plane''' in the theory of [[quadratic form]]s. A common instance has ''F'' = [[real number]]s in which case
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| <math>\scriptstyle \lbrace x \isin V : q(x)\ =\ \text{nonzero constant} \rbrace </math> and
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| <math>\scriptstyle \lbrace x \isin V : r(x)\ =\ \text{nonzero constant} \rbrace </math> are [[hyperbola]]s. In particular, <math>\scriptstyle \lbrace x \isin V : r(x) = 1 \rbrace </math> is the [[unit hyperbola]]. The notation
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| <math>\scriptstyle \langle 1 \rangle \oplus \langle -1 \rangle </math> has been used by Milnor and Huseman<ref>Milnor & Husemoller (1973) page 9</ref> for the hyperbolic plane as the signs of the terms of the [[polynomial|bivariate polynomial]] ''r'' are exhibited.
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| ==Split quadratic space==
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| A space with quadratic form is '''split''' (or '''metabolic''') if there is a subspace which is equal to its own [[orthogonal complement]]: equivalently, the index of isotropy is equal to half the dimension.<ref name=MH57>Milnor & Husemoller (1973) p.57</ref> The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.<ref>Milnor & Husemoller (1973) pp.12–13</ref>
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| == Relation with classification of quadratic forms ==
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| From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field ''F'', classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By [[Witt's decomposition theorem]], every inner product space over a field is an [[orthogonal direct sum]] of a split space and an anisotropic space.<ref>Milnor & Husemoller (1973) p.56</ref>
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| ==Field theory==
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| * If ''F'' is an [[algebraically closed]] field, for example, the field of [[complex numbers]], and (''V'',''q'') is a quadratic space of dimension at least two, then it is isotropic.
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| * If ''F'' is a [[finite field]] and (''V'',''q'') is a quadratic space of dimension at least three, then it is isotropic.
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| * If ''F'' is the field ''Q''<sub>''p''</sub> of [[p-adic number]]s and (''V'',''q'') is a quadratic space of dimension at least five, then it is isotropic.
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| == See also ==
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| *[[Null vector]]
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| *[[Witt group]]
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| *[[Witt ring (forms)]]
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| *[[Witt's theorem]]
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| *[[Symmetric bilinear form]]
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| *[[Universal quadratic form]]
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| == References ==
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| {{reflist}}
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| * Pete L. Clark, [http://www.math.miami.edu/~armstrong/685fa12/pete_clark.pdf Quadratic forms chapter I: Witts theory] from [[University of Miami]] in [[Coral Gables, Florida]].
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| * [[Tsit Yuen Lam]] (1973) ''Algebraic Theory of Quadratic Forms'', §1.3 Hyperbolic plane and hyperbolic spaces, [[W. A. Benjamin]].
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| * Tsit Yuen Lam (2005) ''Introduction to Quadratic Forms over Fields'', [[American Mathematical Society]] ISBN 0-8218-1095-2 .
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| * {{cite book | first1=J. | last1=Milnor | author1-link=John Milnor| first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 }}
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| * {{cite book | first=O.T | last=O'Meara | authorlink=O. Timothy O'Meara | year=1963 | title=Introduction to Quadratic Forms | page=94 §42D Isotropy | publisher=[[Springer-Verlag]] | isbn=3-540-66564-1 }}
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| * {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=A Course in Arithmetic | series=[[Graduate Texts in Mathematics]] | volume=7 | publisher=[[Springer-Verlag]] | year=2000 | origyear=1973 | edition=reprint of 3rd | series=Classics in mathematics | isbn=0-387-90040-3 | zbl=1034.11003 }}
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| [[Category:Quadratic forms]]
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| [[Category:Bilinear forms]]
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