Positive Definite Quadratic Form

A quadratic form is said to be positive definite if for . A real quadratic form in variables is positive definite iff its canonical form is

(1)

A binary quadratic form

(2)

of two real variables is positive definite if it is for any , therefore if and the binary quadratic form discriminant . A binary quadratic form is positive definite if there exist nonzero and such that

(3)

(Le Lionnais 1983).

The positive definite quadratic form

(4)

is said to be reduced if , , and if or . Under the action of the general linear group , i.e., under the set of linear transformations of coordinates with integer coefficients and determinant , there exists a unique reduced positive definite binary quadratic form equivalent to any given one.

There exists a one-to-one correspondence between the set of reduced quadratic forms with fundamental discriminant and the set of classes of fractional ideals of the unique quadratic field with discriminant . Let be a reduced positive definite binary quadratic form with fundamental discriminant , and consider the map which maps the form to the ideal class containing the ideal . Then this map is one-to-one and onto. Thus, the class number of the imaginary quadratic field is equal to the number of reduced binary quadratic forms of discriminant , which can be easily computed by systematically constructing all binary quadratic forms of discriminant by looping over the coefficients and . The third coefficient is then determined by , , and .

A quadratic form is positive definite iff every eigenvalue of is positive. A quadratic form with a Hermitian matrix is positive definite if all the principal minors in the top-left corner of are positive, in other words

			(5)
			(6)
			(7)