Binary quadratic form: Difference between revisions

Revision as of 18:41, 13 March 2017

Revision as of 20:55, 13 March 2017

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For example, with <math>f= x^2 + 4xy + 2y^2</math> and <math>\alpha = -3</math>, <math>\beta = 2</math>, <math>\gamma = 1</math>, and <math>\delta = -1</math>, we find that ''f'' is equivalent to <math>g = (-3x+2y)^2 + 4(-3x+2y)(x-y)+2(x-y)^2</math>, which simplifies to <math>-x^2+4xy-2y^2</math>.

The above equivalence conditions define an [[equivalence relation]] on the set of integral quadratic forms. It follows that the quadratic forms are [[partition of a set|partition]]ed{{dn|date=March 2017}} into equivalence classes, called '''classes''' of quadratic forms. A '''class invariant''' can mean either a function defined on equivalence classes of forms or a property shared by all forms in the same class.

Lagrange used a different notion of equivalence, in which the second condition is replaced by <math> \alpha \delta - \beta \gamma = \pm 1</math>. Since Gauss it has been recognized that this definition is inferior to that given above. If there is a need to distinguish, sometimes forms are called '''properly equivalent''' using the definition above and '''improperly equivalent''' if they are equivalent in Lagrange's sense.