Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of 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 "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of V.

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 Q and its associated symmetric bilinear form B are related by the following equations:

Q

(

x

)

=

B

(

x

,

x

)

B

(

x

,

y

)

=

B

(

y

,

x

)

=

1

2

[

Q

(

x

+

y

)

−

Q

(

x

)

−

Q

(

y

)

]

.

{\displaystyle {\begin{aligned}Q(x)&=B(x,x)\\B(x,y)&=B(y,x)={\tfrac {1}{2}}[Q(x+y)-Q(x)-Q(y)]~.\end{aligned}}}

The latter formula (the polarization identity) arises from expanding

Q

(

x

+

y

)

=

B

(

x

+

y

,

x

+

y

)

.

{\displaystyle \;Q(x+y)=B(x+y,x+y)~.}

Examples

As an example, let

V

=

R

2

{\displaystyle V=\mathbb {R} ^{2}}

, and consider the quadratic form

Q

(

x

)

=

c

1

x

1

2

+

c

2

x

2

2

{\displaystyle Q(x)=c_{1}{x_{1}}^{2}+c_{2}{x_{2}}^{2}}

where

x

=

[

x

1

,

x

2

]

∈

V

{\displaystyle ~x=[x_{1},x_{2}]\in V}

and c1 and c2 are constants.