Bilinear Form


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bilinear form

[¦bī‚lin·ē·ər ′fȯrm]
(mathematics)
A polynomial of the second degree which is homogeneous of the first degree in each of two sets of variables; thus, it is a sum of terms of the form aijxiyj, where x1, … , xm and y1, … , yn are two sets of variables and the aij are constants.
More generally, a mapping ƒ(x, y) from E × F into R, where R is a commutative ring and E × F is the Cartesian product of two modules E and F over R, such that for each x in E the function which takes y into ƒ(x, y) is linear, and for each y in F the function which takes x into ƒ(x, y) is linear.

Bilinear Form

 

a form—that is, a homogeneous polynomial—of the second degree from two groups of variables x1, x2, . . . , xn and y1, y2, . . . , yn nof the form

For example, axy is a bilinear form of the variables x and y, and a11x1y1 + a21x2y1 + a22x2y2 is a bilinear form of the variables x1, x2, and y1, y2. A bilinear form is a particular type of quadratic form.

References in periodicals archive ?
Let g be a Lie algebra, [pi] a representation of g on U, U a g-submodule of Hom(U, C), B a non-degenerate invariant bilinear form on g.
H3) There exists a G-invariant non-degenerate bilinear form on g.
Once the bilinear form of the NLEE was given, multisoliton solutions can be derived through the truncated formal perturbation expansion at different levels [40].
q] a finite field, together with the totally isotropic projective lines of a non-degenerate alternating bilinear form on PG(3, q).
a symmetric and a nonsymmetric bilinear form of the finite element and the finite volume formulations, respectively.
Once the error in the quantity of interest has been determined in terms of the bilinear form b, a sharp upper bound for [absolute value of L(e)] that depends upon the mesh parameters (element size h and order of approximation p) only locally, must be obtained.
By definition [PHI] is an antisymmetric bilinear form and can therefore be expressed in the coordinates ([u.
Define on this space the bilinear form by the formula
The equation extrapolate future input-output behaviours from past input-output data, which are in bilinear form because there are products of control inputs in the construction of the data matrices [W.
Our mechanism of constructing subspaces of V([Lambda]), on which, hopefully, the bilinear form <[Phi], [Psi]> is negative, is based on the existence of one-parameter groups of diffeomorphisms of [R.
where the bilinear form [mathematical expression not reproducible] and the linear function [mathematical expression not reproducible] are defined on elements.