Bilinear Form

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 forma_ijx_iy_j, wherex₁, … ,x_mandy₁, … ,y_nare two sets of variables and thea_ijare constants.

More generally, a mapping ƒ(x, y) fromE×FintoR, whereRis a commutative ring andE×Fis the Cartesian product of two modulesEandFoverR, such that for eachxinEthe function which takesyinto ƒ(x, y) is linear, and for eachyinFthe function which takesxinto ƒ(x, y) is linear.

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Bilinear Form 

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