Definite quadratic form

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 nonzero 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 "always nonnegative" and "always nonpositive", respectively. In other words, it may take on zero values.
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.
Associated symmetric bilinear form.
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. 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:
The latter formula arises from expanding formula_2.
Examples.
As an example, let formula_3, and consider the quadratic form
where formula_5 and and are constants. If and , the quadratic form is positive-definite, so "Q" evaluates to a positive number whenever formula_6 If one of the constants is positive and the other is 0, then is positive semidefinite and always evaluates to either 0 or a positive number. If and , or vice versa, then is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If and , the quadratic form is negative-definite and always evaluates to a negative number whenever formula_6 And if one of the constants is negative and the other is 0, then is negative semidefinite and always evaluates to either 0 or a negative number.
In general a quadratic form in two variables will also involve a cross-product term in "x"1"x"2:
This quadratic form is positive-definite if formula_9 and formula_10 negative-definite if formula_11 and formula_10 and indefinite if formula_13 It is positive or negative semidefinite if formula_14 with the sign of the semidefiniteness coinciding with the sign of formula_15
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 formula_16
The square of the Euclidean norm in "n"-dimensional space, the most commonly used measure of distance, is
In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the formula_18 axis and the formula_19 axis.
Matrix form.
A quadratic form can be written in terms of matrices as
where "x" is any "n"×1 Cartesian vector formula_21 in which not all elements are 0, superscript T denotes a transpose, and "A" is an "n"×"n" symmetric matrix. If "A" is diagonal this is equivalent to a non-matrix form containing solely terms involving squared variables; but if "A" has any non-zero off-diagonal elements, the non-matrix form will also contain some terms involving products of two different variables.
Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to the same property of "A", which can be checked by considering all eigenvalues of "A" or by checking the signs of all of its principal minors.
Optimization.
Definite quadratic forms lend themselves readily to optimization problems. Suppose the matrix quadratic form is augmented with linear terms, as
where "b" is an "n"×1 vector of constants. The first-order conditions for a maximum or minimum are found by setting the matrix derivative to the zero vector:
giving
assuming "A" is nonsingular. If the quadratic form, and hence "A", 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.