In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every non-zero vector of . According to that sign, the quadratic form is called positive-definite or negative-definite.
A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of .
An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form.
More generally, these definitions apply to any vector space over an ordered field.[1]
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.[2] A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form and its associated symmetric bilinear form are related by the following equations:
\begin{align} Q(x)&=B(x,x)\\ B(x,y)&=B(y,x)=\tfrac{1}{2}[Q(x+y)-Q(x)-Q(y)]~. \end{align}
The latter formula arises from expanding
Q(x+y)=B(x+y,x+y)~.
As an example, let
V=R2
Q(x)=c1{x
2 | |
1} |
+c2{x
2 | |
2} |
~x=[x1,x2]\inV
[x1,x2] ≠ [0,0]~.
[x1,x2] ≠ [0,0]~.
In general a quadratic form in two variables will also involve a cross-product term in ·:
Q(x)=c1
2 | |
{x | |
1} |
+c2
2 | |
{x | |
2} |
+2c3x1x2~.
This quadratic form is positive-definite if
c1>0
c1c2-
2 | |
{c | |
3} |
>0 ,
c1<0
c1c2-
2 | |
{c | |
3} |
>0 ,
c1c2-
2 | |
{c | |
3} |
<0~.
c1c2-
2 | |
{c | |
3} |
=0 ,
c1~.
This bivariate quadratic form appears in the context of conic sections centered on the origin. If the general quadratic form above is equated to 0, the resulting equation is that of an ellipse if the quadratic form is positive or negative-definite, a hyperbola if it is indefinite, and a parabola if
c1c2-
2=0 | |
{c | |
3} |
~.
The square of the Euclidean norm in -dimensional space, the most commonly used measure of distance, is
2 | |
{x | |
1} |
+ … +
2 | |
{x | |
n} |
~.
In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the
x1
x2
A quadratic form can be written in terms of matrices as
xTAx
[x1, … ,
T | |
x | |
n] |
Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to the same property of , which can be checked by considering all eigenvalues of or by checking the signs of all of its principal minors.
Definite quadratic forms lend themselves readily to optimization problems. Suppose the matrix quadratic form is augmented with linear terms, as
xTAx+bTx ,
where is an ×1 vector of constants. The first-order conditions for a maximum or minimum are found by setting the matrix derivative to the zero vector:
2Ax+b=\vec0 ,
giving
x=-\tfrac{1}{2}A-1b ,
assuming is nonsingular. If the quadratic form, and hence, is positive-definite, the second-order conditions for a minimum are met at this point. If the quadratic form is negative-definite, the second-order conditions for a maximum are met.
An important example of such an optimization arises in multiple regression, in which a vector of estimated parameters is sought which minimizes the sum of squared deviations from a perfect fit within the dataset.