Definite quadratic form: Difference between revisions

Revision as of 21:48, 10 October 2017

Revision as of 18:37, 17 July 2018

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In [[mathematics]], a '''definite quadratic form''' is a [[quadratic form]] over some [[Real number|real]] [[vector space]] {{math|''V''}} that has the same [[positive and negative numbers|sign]] (always positive or always negative) for every nonzero vector of {{math|''V''}}. According to that sign, the quadratic form is called '''positive -definite''' or '''negative -definite'''.

A '''semidefinite''' (or '''semi-definite''') quadratic form is defined in the same way, except that "positive" and "negative" are replaced by "not negative" and "not positive", respectively. An '''indefinite''' quadratic form is one that takes on both positive and negative values.

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:<math>Q(x)=c_1{x_1}^2+c_2{x_2}^2 </math>

where {{math|1=''x'' = (''x''₁, ''x''₂)}} and {{math|''c''₁}} and {{math|''c''₂}} are constants. If {{math|1=''c''₁ > 0}} and {{math|1=''c''₂ > 0}}, the quadratic form {{math|''Q''}} 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 {{math|''Q''}} is positive semidefinite and always evaluates to either 0 or a positive number. If {{math|1=''c''₁ > 0}} and {{math|1=''c''₂ < 0}}, or vice versa, then {{math|''Q''}} is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If {{math|1=''c''₁ < 0}} and {{math|1=''c''₂ < 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 {{math|''Q''}} 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:

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:<math>Q(x)=c_1{x_1}^2+c_2{x_2}^2+2c_3x_1x_2.</math>

This quadratic form is positive -definite if <math>c_1>0</math> and <math>c_1c_2-{c_3}^2>0,</math> negative -definite if <math>c_1<0</math> and <math>c_1c_2-{c_3}^2>0,</math> and indefinite if <math>c_1c_2-{c_3}^2<0.</math> It is positive or negative semidefinite if <math>c_1c_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 section]]s 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_1c_2-{c_3}^2=0.</math>

The square of the [[Euclidean norm]] in ''n''-dimensional space, the most commonly used measure of distance, is

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where ''x'' is any ''n''×1 [[Euclidean vector#In Cartesian space|Cartesian vector]] <math>(x_1, \cdots , x_n)^T </math> in which not all elements are 0, superscript ^T denotes a [[transpose]], and ''A'' is an ''n''×''n'' [[symmetric matrix]]. If ''A'' is [[diagonal matrix|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 [[positive -definite matrix|the same property of ''A'']], which can be checked by considering all [[eigenvalue]]s of ''A'' or by checking the signs of all of its [[principal minor]]s.

==Optimization==

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:<math>x=-A^{-1}b</math>

assuming ''A'' is [[nonsingular matrix|nonsingular]]. If the quadratic form, and hence ''A'', is positive -definite, the [[second partial derivative test|second-order condition]]s 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.

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==See also==

*[[Anisotropic quadratic form]]

*[[Positive -definite function]]

*[[Positive -definite matrix]]

*[[Polarization identity]]