1911 Encyclopædia Britannica/Number/Bilinear Forms

3373321911 Encyclopædia Britannica, Volume 19 — - Number Bilinear Forms

40. Bilinear Forms.—A bilinear form means an expression of the type ; the most important case is when , and only this will be considered here. The invariants of a form are its determinant and the elementary factors thereof. Two bilinear forms are equivalent when each can be transformed into the other by linear integral substitutions . Every bilinear form is equivalent to a reduced form , and , unless . In order that two forms may be equivalent it is necessary and sufficient that their invariants should be the same. Moreover, if and , and if the invariants of the forms are the same for all values of , we shall have , and the transformation of one form to the other may be effected by a substitution which does not involve . The theory of bilinear forms practically includes that of quadratic forms, if we suppose to be cogredient variables. Kronecker has developed the case when , 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 is , where is or according as is or is not a square, and the symbols have the meaning previously assigned to them (§ 39).