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 ?
In order to estimate the convective term in the bilinear form a, we integrate by parts and obtain
h ]be the linear operator representing the bilinear form [a.
a symmetric and a nonsymmetric bilinear form of the finite element and the finite volume formulations, respectively.
Given a Coxeter system (W, S), we define a bilinear form B as follows:
By definition [PHI] is an antisymmetric bilinear form and can therefore be expressed in the coordinates ([u.
which is a time dependent bilinear form representing the linear combination of various orders of derivatives.
The power expended by the driving force can be written as the general bilinear form
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.
This proof is based on the study of a bilinear form that already appeared in [4] but was not explored yet.
We extend U complex linearly to a bilinear form on ([h.