1911 Encyclopædia Britannica/Number/Bilinear Forms

40. Bilinear Forms.&mdash;A bilinear form means an expression of the type $$\Sigma a_{ik}x_iy_k\, (i = 1, 2,\dots m; k = 1, 2,\dots n)$$; the most important case is when $$m = n$$, and only this will be considered here. The invariants of a form are its determinant $$[a_{nn}]$$ and the elementary factors thereof. Two bilinear forms are equivalent when each can be transformed into the other by linear integral substitutions $$x'=\Sigma\alpha x, y'=\Sigma\beta y$$. Every bilinear form is equivalent to a reduced form $$\overset{r}{\underset{1}{\Sigma}} e_i x_i y_i$$, and $$r=n$$, unless $$[a_{nn}]=0$$. In order that two forms may be equivalent it is necessary and sufficient that their invariants should be the same. Moreover, if $$a \sim b$$ and $$c \sim d$$, and if the invariants of the forms $$a+\lambda c, b+\lambda d$$ are the same for all values of $$\lambda$$, we shall have $$a+\lambda c \sim b+\lambda d$$, and the transformation of one form to the other may be effected by a substitution which does not involve $$\lambda$$. The theory of bilinear forms practically includes that of quadratic forms, if we suppose $$x_i, y_i$$ to be cogredient variables. Kronecker has developed the case when $$n=2$$, and deduced various class-relations for quadratic forms in a manner resembling that of Liouville. So far as the bilinear forms are concerned, the main result is that the number of classes for the positive determinant $$a_{11}a_{22}-a_{12}a_{21}=\Delta$$ is $$12\{\Phi(\Delta)+\Psi(\Delta)\}+2\epsilon$$, where $$\epsilon$$ is $$1$$ or $$0$$ according as $$\Delta$$ is or is not a square, and the symbols $$\Phi, \Psi$$ have the meaning previously assigned to them (&sect; 39).