In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements...
22 KB (2,700 words) - 23:21, 30 May 2024
specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space...
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ordered field. Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite...
7 KB (1,202 words) - 18:41, 10 June 2022
In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the...
8 KB (1,511 words) - 05:02, 19 May 2023
form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is...
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Thus, bq is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form q ( x ) = b ( x , x ) ,...
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In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each...
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Transpose (section Transpose of a bilinear form)
defines a bilinear form B : X × X → F, with the relation B(x, y) = u(x)(y). By defining the transpose of this bilinear form as the bilinear form tB defined...
20 KB (2,525 words) - 12:30, 8 May 2024
Change of basis (section Bilinear forms)
of the bilinear form on the new basis is P T B P . {\displaystyle P^{\mathsf {T}}\mathbf {B} P.} A symmetric bilinear form is a bilinear form B such that...
17 KB (3,077 words) - 06:23, 31 August 2024
Trace (linear algebra) (section Bilinear forms)
{X} } is called the Killing form, which is used for the classification of Lie algebras. The trace defines a bilinear form: ( X , Y ) ↦ tr ( X Y ) ....
36 KB (5,385 words) - 17:42, 20 July 2024