Definite quadratic form: Difference between revisions

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{{Short description|Type of homogeneous polynomial of degree 2}}

In [[mathematics]], a '''definite quadratic form''' is a [[quadratic form]] over some [[Real number|real]] [[vector space]] {{mathmvar|''V''}} that has the same [[positive and negative numbers|sign]] (always positive or always negative) for every non-zero 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 much the same way, except that "always positive" and "always negative" are replaced by "alwaysnever nonnegativenegative" and "alwaysnever nonpositivepositive", respectively. In other words, it may take on zero values for some non-zero vectors of {{mathmvar|''V''}}.

An '''indefinite''' quadratic form takes on both positive and negative values and is called an [[isotropic quadratic form]].

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==Associated symmetric bilinear form==

Quadratic forms correspond one-to-one to [[symmetric bilinear form]]s over the same space.<ref>This is true only over a field of [[characteristic (algebra)|characteristic]] other than 2, but here we consider only [[ordered field]]s, which necessarily have characteristic 0.</ref> A symmetric bilinear form is also described as '''definite''', '''semidefinite''', etc. according to its associated quadratic form. A quadratic form {{mathmvar|''Q''}} and its associated symmetric bilinear form {{mathmvar|''B''}} are related by the following equations:

:<math>\begin{align}

Q(x) &= B(x, x) \\

B(x,y) &= B(y,x) = \fractfrac{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|1=''>~ x'' = (''x''₁[x_1, ''x''₂)}} <math>x_2] \in V </math> and {{mathmvar|''c''<}}{{sub>|1</sub>}} and {{mathmvar|''c''<}}{{sub>|2</sub>}} are constants. If {{mathnobr| {{mvar|1=''c''<}}{{sub>|1</sub>}} > 0 }} and {{mathnobr| {{mvar|1=''c''<}}{{sub>|2</sub>}} > 0 ,}}, the quadratic form {{mathmvar|''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 {{mathmvar|''Q''}} is positive semidefinite and always evaluates to either 0 or a positive number. If {{mathnobr| {{mvar|1=''c''<}}{{sub>|1</sub>}} > 0 }} and {{mathnobr| {{mvar|1=''c''<}}{{sub>|2</sub>}} < 0 ,}}, or vice versa, then {{mathmvar|''Q''}} is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If {{mathnobr| {{mvar|1=''c''<}}{{sub>|1</sub>}} < 0 }} and {{mathnobr| {{mvar|1=''c''<}}{{sub>|2</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 {{mathmvar|''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 ''{{mvar|x''<}}{{sub>|1</sub>''}}·{{mvar|x''<}}{{sub>|2</sub>}}:

:<math> Q(x) = c_1 {x_1}^2 + c_2 {x_2}^2 +2c_3x_1x_2 2 c_3 x_1 x_2 ~.</math>

This quadratic form is positive-definite if <math>\; c_1 > 0 \;</math> and <math>c_1c_2\, c_1 c_2 - {c_3}^2 > 0 \;,</math> negative-definite if <math>\; c_1 < 0 \;</math> and <math>c_1c_2\, c_1 c_2 - {c_3}^2 > 0 \;,</math> and indefinite if <math>c_1c_2\; c_1 c_2 - {c_3}^2 < 0 ~.</math> It is positive or negative semidefinite if <math>c_1c_2\; 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 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_1 c_2 - {c_3}^2=0 ~.</math>

The square of the [[Euclidean norm]] in ''{{mvar|n''}}-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.

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A quadratic form can be written in terms of [[matrix (mathematics)|matrices]] as

:<math>x^\textmathsf{T} A \, x</math>

where ''{{mvar|x''}} is any ''{{mvar|n''}}×1 [[Euclidean vector#In Cartesian space|Cartesian vector]] <math>(\; [x_1, \cdots , x_n)]^\textmathsf{T} \;</math> in which notat allleast elementsone areelement is not 0,; superscript{{mvar|A}} ^Tis denotesan a{{mvar| n × n }} [[transposesymmetric matrix]],; and ''A''superscript is{{sup|T}} andenotes ''n''×''n''a [[symmetric matrix transpose]]. If ''{{mvar|A''}} is [[diagonal matrix|diagonal]] this is equivalent to a non-matrix form containing solely terms involving squared variables; but if ''{{mvar|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 ''{{mvar|A''}}]], which can be checked by considering all [[eigenvalue]]s of ''{{mvar|A''}} or by checking the signs of all of its [[principal minor]]s.

==Optimization==

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Definite quadratic forms lend themselves readily to [[optimization]] problems. Suppose the matrix quadratic form is augmented with linear terms, as

:<math>x^\textmathsf{T} A \, x + 2 b^\textmathsf{T} x \;,</math>

where ''{{mvar|b''}} is an ''{{mvar|n''}}×1 vector of constants. The [[first-order condition]]s for a maximum or minimum are found by setting the [[matrix derivative]] to the zero vector:

:<math>2Ax 2 A \, x +2b b = \vec 0 \;,</math>

giving

:<math> x = -\tfrac{1}{2}\,A^{-1}b \;,</math>

assuming ''{{mvar|A''}} is [[nonsingular matrix|nonsingular]]. If the quadratic form, and hence ''{{mvar|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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==References==

*{{cite book

| last=Kitaoka | first=Yoshiyuki

| firstyear=Yoshiyuki1993

| title=Arithmetic of quadratic forms

| series=Cambridge Tracts in Mathematics

| volume=106

| publisher=Cambridge University Press

| year=1993

| isbn=0-521-40475-4

| zbl=0785.11021

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}}.

*{{cite book

| first1=J. | last1=Milnor | author1-link=John Milnor

| first2=D. | last2=Husemoller

| last1=Milnor

| year=19931973

| author1-link=John Milnor

| first2=D.

| last2=Husemoller

| title=Symmetric Bilinear Forms

| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]

| volume=73

| publisher=Springer

| year=1973

| isbn=3-540-06009-X

| zbl=0292.10016