# 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
Then there is a sequence of Laurent polynomials [R.sub.0], [R.sub.1],..., [R.sub.2m] that are orthonormal with respect to the bilinear form
Schwabik in [3] considers a bilinear form, defines a Stieltjes type integral, and performs a study about it including [4]; following his ideas we give integration by parts theorem involving a bilinear operator and, through it, we prove a representation theorem for the space of Henstock vectorvalued functions.
with the discrete bilinear form [a.sup.swip.sub.h] yet to be designed, has a unique solution, since the right-hand side (2) [[integral].sub.[OMEGA]] f[v.sub.h]dx is correctly defined for f [member of] [L.sup.1]([OMEGA]) and the bilinear form [a.sup.swip.sub.h] is consistent.
A bilinear form B : (M [cross product] M, [mu] [cross product] [mu]) [right arrow] (k, id) in [??]([M.sub.k]) is called surgenerate if there exists an isomorphism [phi] : (M, [mu]) [right arrow] ([M.sup.*], [([[mu].sup.*]).sup.-1]) in [??]([M.sub.k]) given by
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.
Here [alpha] is chosen to guarantee the [H.sup.1]-coercivity of the bilinear form in the second equation.
Consider a(x, x) is the symmetric bilinear form on [H.sup.1.sub.0] x [H.sup.1.sub.0] defined by
Let us introduce a bilinear form a(u, v): H x H [right arrow] R and a linear form [b.sub.([eta])(v)] : H [right arrow] R defined as
The bilinear form p defined on the Lie-Yamaguti superalgebra T = [T.sub.0] direct sum] [T.sub.1] by
Tang in [27] obtained its Pfaffian solution and extended Pfaffian solutions with the aid of the Hirota bilinear form. Su et al.
Assume that the bilinear form k(*, *) satisfies the inf-sup condition
(H3) There exists a G-invariant non-degenerate bilinear form on g.

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