This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.
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*PM : hermitian form over a division ring, id = 7626-- WP guess : hermitian form over a division ring-- Status:
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*PM : hermitian form over a division ring, id = 7626-- WP guess : hermitian form over a division ring-- Status:
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While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a " positive-definite Hermitian form ".
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On an almost K�hler manifold, one can write this decomposition as, where is the Hermitian form, is the Riemannian metric, is the almost symplectic structure.
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The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces.
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A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on " U " ?" U ".
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There is the related notion of "'?-symmetric forms "', which generalizes symmetric forms, skew-symmetric forms ( = symplectic forms ), Hermitian forms, and skew-Hermitian forms.
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There is the related notion of "'?-symmetric forms "', which generalizes symmetric forms, skew-symmetric forms ( = symplectic forms ), Hermitian forms, and skew-Hermitian forms.
20.
Given a Hermitian form ? on a complex vector space " V ", the unitary group U ( ? ) is the group of transforms that preserve the form : the transform " M " such that for all.
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