Английская Википедия:Definite quadratic form
Шаблон:Short description In mathematics, a definite quadratic form is a quadratic form over some real vector space Шаблон:Mvar that has the same sign (always positive or always negative) for every non-zero vector of Шаблон:Math. 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 Шаблон:Mvar.
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]
Associated symmetric bilinear form
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 Шаблон:Mvar and its associated symmetric bilinear form Шаблон:Mvar are related by the following equations:
- <math>\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}</math>
The latter formula arises from expanding <math>\; Q(x+y) = B(x+y,x+y) ~.</math>
Examples
As an example, let <math>V = \mathbb{R}^2 </math>, and consider the quadratic form
- <math> Q(x) = c_1{x_1}^2 + c_2{x_2}^2 </math>
where <math>~ x = [x_1, x_2] \in V </math> and Шаблон:MvarШаблон:Sub and Шаблон:MvarШаблон:Sub are constants. If Шаблон:MvarШаблон:Sub > 0 and Шаблон:MvarШаблон:Sub > 0 , the quadratic form Шаблон:Mvar is positive-definite, so Q evaluates to a positive number whenever <math>\; [x_1,x_2] \neq [0,0] ~.</math> If one of the constants is positive and the other is 0, then Шаблон:Mvar is positive semidefinite and always evaluates to either 0 or a positive number. If Шаблон:MvarШаблон:Sub > 0 and Шаблон:MvarШаблон:Sub < 0 , or vice versa, then Шаблон:Mvar is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If Шаблон:MvarШаблон:Sub < 0 and Шаблон:MvarШаблон:Sub < 0 , the quadratic form is negative-definite and always evaluates to a negative number whenever <math>\; [x_1,x_2] \neq [0,0] ~.</math> And if one of the constants is negative and the other is 0, then Шаблон:Mvar 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 Шаблон:MvarШаблон:Sub·Шаблон:MvarШаблон:Sub:
- <math> Q(x) = c_1 {x_1}^2 + c_2 {x_2}^2 + 2 c_3 x_1 x_2 ~.</math>
This quadratic form is positive-definite if <math>\; c_1 > 0 \;</math> and <math>\, c_1 c_2 - {c_3}^2 > 0 \;,</math> negative-definite if <math>\; c_1 < 0 \;</math> and <math>\, c_1 c_2 - {c_3}^2 > 0 \;,</math> and indefinite if <math>\; c_1 c_2 - {c_3}^2 < 0 ~.</math> It is positive or negative semidefinite if <math>\; c_1 c_2 - {c_3}^2 = 0 \;,</math> with the sign of the semidefiniteness coinciding with the sign of <math>\; c_1 ~.</math>
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 <math>\; c_1 c_2 - {c_3}^2=0 ~.</math>
The square of the Euclidean norm in Шаблон:Mvar-dimensional space, the most commonly used measure of distance, is
- <math> {x_1}^2 +\cdots + {x_n}^2 ~.</math>
In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the <math>x_1</math> axis and the <math>x_2</math> axis.
Matrix form
A quadratic form can be written in terms of matrices as
- <math>x^\mathsf{T} A \, x</math>
where Шаблон:Mvar is any Шаблон:Mvar×1 Cartesian vector <math>\; [x_1, \cdots , x_n]^\mathsf{T} \;</math> in which at least one element is not 0; Шаблон:Mvar is an Шаблон:Mvar symmetric matrix; and superscript Шаблон:Sup denotes a matrix transpose. If Шаблон:Mvar is diagonal this is equivalent to a non-matrix form containing solely terms involving squared variables; but if Шаблон:Mvar 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 [[positive-definite matrix|the same property of Шаблон:Mvar]], which can be checked by considering all eigenvalues of Шаблон:Mvar 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
- <math>x^\mathsf{T} A \, x + b^\mathsf{T} x \;,</math>
where Шаблон:Mvar is an Шаблон:Mvar×1 vector of constants. The first-order conditions for a maximum or minimum are found by setting the matrix derivative to the zero vector:
- <math> 2 A \, x + b = \vec 0 \;,</math>
giving
- <math> x = -\tfrac{1}{2}\,A^{-1}b \;,</math>
assuming Шаблон:Mvar is nonsingular. If the quadratic form, and hence Шаблон:Mvar, 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.
See also
Notes
References
- ↑ Шаблон:Harvnb.
- ↑ This is true only over a field of characteristic other than 2, but here we consider only ordered fields, which necessarily have characteristic 0.
