TOPICS
Search

Positive Definite Quadratic Form


A quadratic form Q(z) is said to be positive definite if Q(z)>0 for z!=0. A real quadratic form in n variables is positive definite iff its canonical form is

 Q(z)=z_1^2+z_2^2+...+z_n^2.
(1)

A binary quadratic form

 F(x,y)=ax^2+bxy+cy^2
(2)

of two real variables is positive definite if it is >0 for any (x,y)!=(0,0), therefore if a>0 and the binary quadratic form discriminant d=4ac-b^2>0. A binary quadratic form is positive definite if there exist nonzero x and y such that

 (ax^2+bxy+cy^2)^2<=4/3|4ac-b^2|
(3)

(Le Lionnais 1983).

The positive definite quadratic form

 F=<a,b,c>=ax^2+bxy+cy^2
(4)

is said to be reduced if |b|<=a<=c, c>0, and b>0 if a=|b| or a=c. Under the action of the general linear group GL(2,Z), i.e., under the set of linear transformations of coordinates (x,y) with integer coefficients and determinant +/-1, 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 D<0 and the set of classes of fractional ideals of the unique quadratic field with discriminant D. Let F=<a,b,c> be a reduced positive definite binary quadratic form with fundamental discriminant D<0, and consider the map phi_(FI) which maps the form F to the ideal class containing the ideal (a,(-b+sqrt(D))/2)). Then this map is one-to-one and onto. Thus, the class number of the imaginary quadratic field Q(sqrt(D)) is equal to the number of reduced binary quadratic forms of discriminant D, which can be easily computed by systematically constructing all binary quadratic forms of discriminant D by looping over the coefficients a and b. The third coefficient c is then determined by a, b, and D.

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

a_(11)>0
(5)
|a_(11) a_(12); a_(21) a_(22)|>0
(6)
|a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)|>0.
(7)

See also

Binary Quadratic Form, Indefinite Quadratic Form, Lyapunov's First Theorem, Positive Semidefinite Quadratic Form, Quadratic Form

Portions of this entry contributed by David Terr

Explore with Wolfram|Alpha

References

Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, pp. 221-224, 1993.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1106, 2000.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 38, 1983.

Referenced on Wolfram|Alpha

Positive Definite Quadratic Form

Cite this as:

Terr, David and Weisstein, Eric W. "Positive Definite Quadratic Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html

Subject classifications