# Factoid Details

### ‘A (bilinear form) can be symmetric.’

[⟨det (nn bilinear.n form.n)⟩ symmetric.a]

In Wikipedia:

Every element “x” of “g” defines the adjoint endomorphism “ad” (“x”) (also written as “ad ’x”) of “g” with the help of the Lie bracket, as Now, supposing “g” is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form with values in “K”, the Killing form’ “on” g”.
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In Wikipedia:

We say that two symmetric bilinear forms are equivalent’ “if one can be obtained from the other by adding zero or more copies of a hyperbolic plane (the non – degenerate two – dimensional symmetric bilinear form with a norm 0 vector).
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Thus “B” is a positive definite symmetric bilinear form, in other words, an inner product on “V”.
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A map [Formula 1] is a symmetric bilinear form on the space if: The last two axioms only imply linearity in the first argument, but the first immediately implies linearity in the second argument then too.
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Here ““is a nondegenerate skew symmetric bilinear form where F ’” is the field of scalars.
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The associated symmetric bilinear form on “C” (“V”, “Q”) is given by One can check that this reduces to the original bilinear form when restricted to “V”.
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A metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space, which varies smoothly from point to point.
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So, an inner product on a real vector space is a “positive – definite symmetric bilinear form”.
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In mathematics, a quasi – Frobenius Lie algebra’ “over a field [Formula 2] is a Lie algebra equipped with a nondegenerate skew – symmetric bilinear form for all [Formula 8], [Formula 9], [Formula 10] in [Formula 5].
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In Wikipedia:

A GUT model basically consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang – Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra (which is specified by a coupling constant for each factor), a Higgs sector consisting of a number of scalar fields taking on values within real / complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group.
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In Wikipedia:

Taking the standard symmetric bilinear form with orthonormal basis “v ’i”, the map sending a lattice to its dual lattice gives an automorphism with square the identity, giving the permutation that sends each label to its negative modulo “n”.
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For example, let (“V”,) be a finite dimensional real symplectic vector space (so is a nondegenerate skew symmetric bilinear form on “V”).
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An orthogonal basis is one such that where <, > is the symmetric bilinear form associated to “Q”.
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Explicitly, the metric is a symmetric bilinear form on each tangent space of “M” which varies in a smooth (or differentiable) manner from point to point.
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The signature’ “of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric.
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In Wikipedia:

If “B” is any symmetric bilinear form on a vector space, and “Q” is the quadratic form defined by These formulas even apply in the case where the field of scalars has characteristic two, though the left – hand sides are all zero in this case.
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In Wikipedia:

The quadratic form defining the Plcker relation comes from a symmetric bilinear form of signature (3,3).
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The metric “g” is defined to be the real part of “h”: The form “g” is a symmetric bilinear form on “TM” C, the complexified tangent bundle.
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In Wikipedia:

Given any unimodular symmetric bilinear form over the integers, “Q”, there is a simply – connected closed 4 – manifold “M” with intersection form “Q”.
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And 17 more sentences.

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Queried on 2013-06-20.