# 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

*a*, where_{ij}x_{i}y_{j}*x*_{1}, … ,*x*and_{m}*y*_{1}, … ,*y*are two sets of variables and the_{n}*a*are constants._{ij}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 *x*_{1}, *x*_{2}, . . . , *x _{n}* and

*y*

_{1},

*y*

_{2}, . . . ,

*y*nof the form

_{n}For example, *axy* is a bilinear form of the variables *x* and *y*, and *a*_{11}*x*_{1}*y*_{1} + *a*_{21}*x*_{2}*y*_{1} + *a*_{22}*x*_{2}*y*_{2} is a bilinear form of the variables *x*_{1}, *x*_{2}, and *y*_{1}, *y*_{2}. A bilinear form is a particular type of quadratic form.