Bilinear Form

(redirected from Antisymmetric bilinear form)

bilinear form

[¦bī‚lin·ē·ər ′fȯrm]
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 It might be outdated or ideologically biased.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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
By definition [PHI] is an antisymmetric bilinear form and can therefore be expressed in the coordinates ([u.sup.i], [du.sup.i]) as a 2-form [PHI] = [[phi].sub.[ij]] [du.sup.i] [conjunction] [du.sup.j].

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