quadratic form

quadratic form

In this entry, unless otherwise specified,

R

is a

commutative ring

with

multiplicative identity

1

and

M

=

R

⁢

[

X

1

,

…

,

X

n

]

is a

polynomial ring

over

R

in

n

indeterminates

.

Definition

A

homogeneous polynomial

of degree 2 in

M

is called a

quadratic form

(over

R

) in

n

indeterminates. In general, a quadratic form (without specifying

n

) over a ring

R

is a quadratic form in some polynomial ring over

R

.

For example, in

ℤ

⁢

[

X

,

Y

]

,

X

2

-

5

⁢

X

⁢

Y

is a quadratic form, while

Y

3

+

2

⁢

X

⁢

Y

and

X

2

+

Y

2

+

1

are not.

In general, a quadratic form

Q

in

n

-indeterminates looks like

Q

=

a

11

⁢

X

1

2

+

a

12

⁢

X

1

⁢

X

2

+

⋯

+

a

n

,

n

-

1

⁢

X

n

⁢

X

n

-

1

+

a

n

⁢

n

⁢

X

n

2

=

∑

1

≤

i

,

j

≤

n

a

i

⁢

j

⁢

X

i

⁢

X

j

where

a

i

⁢

j

∈

R

.

Letting

𝐗

=

(

X

1

,

…

,

X

n

)

T

, and

𝐀

=

{

a

i

⁢

j

}

the

n

×

n

matrix, then we can rewrite

Q

as

Q

=

𝐗

T

⁢

𝐀𝐗

.

For example, the quadratic form

X

2

-

5

⁢

X

⁢

Y

can be rewritten as

X

2

-

5

⁢

X

⁢

Y

=

(

X

Y

)

⁢

(

1

-

2

-

3

0

)

⁢

(

X

Y

)

.

Now suppose the

characteristic

of

R

,

char

⁡

(

R

)

≠

2

. In fact, suppose that

2

is

invertible

in

R

(its

inverse

denoted by

1

2

). Since

X

i

⁢

X

j

=

X

j

⁢

X

i

, define

b

i

⁢

j

=

1

2

⁢

(

a

i

⁢

j

+

a

j

⁢

i

)

. Then

b

i

⁢

i

=

a

i

⁢

i

and

b

i

⁢

j

=

b

j

⁢

i

. Furthermore, if

𝐁

=

{

b

i

⁢

j

}

, then

𝐁

is a

symmetric matrix

and

Q

=

𝐗

T

⁢

𝐁𝐗

.

Again, in the example of

X

2

-

5

⁢

X

⁢

Y

, over

ℚ

it can be written as

X

2

-

5

⁢

X

⁢

Y

=

(

X

Y

)

⁢

(

1

-

5

2

-

5

2

0

)

⁢

(

X

Y

)

.

However, it is not possible to represent

X

2

-

5

⁢

X

⁢

Y

over

ℤ

by a symmetric matrix.

Evaluating a Quadratic Form

It is not hard to see that, given a quadratic form

Q

in

n

indeterminates, setting one of its indeterminates to

0

gives us another quadratic form, in

(

n

-

1

)

indeterminates. This is an informal way of saying the following:

embed

R

into

N

=

R

⁢

[

X

1

,

…

,

X

n

-

1

]

. Let

ϕ

:

M

→

N

be the (unique)

evaluation homomorphism

of the

embedding

, with

ϕ

⁢

(

X

i

)

=

X

i

for

i

<

n

and

ϕ

⁢

(

X

n

)

=

0

. Then for any quadratic form

Q

∈

M

,

ϕ

⁢

(

Q

)

is a quadratic form in

N

.

In particular, if we take

N

=

R

, and

𝐬

=

(

s

1

,

…

,

s

n

)

with

s

i

∈

R

. Then the evaluation homomorphism

ϕ

at

𝐬

for any quadratic form

Q

∈

M

is called the

evaluation

of

Q

at

𝐬

, and we write it

ϕ

𝐬

⁢

(

Q

)

, or simply

Q

⁢

(

𝐬

)

(since

ϕ

is uniquely determined by

𝐬

). In this way, a quadratic form

Q

can be realized as a

quadratic map

, as follows:

Let

Q

∈

M

be a qudratic form. Take the

direct sum

of

n

copies of

R

and call this

V

. Define a map

q

:

V

→

R

by

q

⁢

(

v

)

=

Q

⁢

(

v

)

. Then

q

is a quadratic map.

Conversely

, if

2

is invertible in

R

(so that

char

⁡

(

R

)

≠

2

is clear), then given a quadratic map

q

:

M

→

R

, one can find a corresponding quadratic form

Q

∈

M

such that

q

⁢

(

v

)

=

Q

⁢

(

v

)

, by setting

a

i

⁢

j

=

1

2

⁢

(

q

⁢

(

e

i

+

e

j

)

-

q

⁢

(

e

i

)

-

q

⁢

(

e

j

)

)

,

where

e

i

and

e

j

are

coordinate vectors

whose coordinates are all

0

except at positions

i

and

j

respectively, where the coordinates are

1

. Then

Q

defined by

𝐗

T

⁢

𝐀𝐗

, where

𝐀

=

{

a

i

⁢

j

}

is the desired quadratic form.

Equivalence of Quadratic Forms

From the above discussion, we shall identify a quadratic form as a quadratic map.

Two quadratic forms

Q

1

and

Q

2

are said to be

if there is an

invertible matrix

M

such that

Q

1

⁢

(

v

)

=

Q

2

⁢

(

M

⁢

v

)

, for all

v

∈

R

n

. The definition of equivalent quadratic forms is well-defined and it is not hard to see that this equivalence is an

equivalence relation

.

In fact, if

𝐀

1

and

𝐀

2

are matrices corresponding to (see the definition

section

) the two equivalent quadratic forms

Q

1

and

Q

2

above, then

𝐀

1

=

M

T

⁢

𝐀

2

⁢

M

.

For example, the quadratic form

X

2

-

Y

2

is

equivalent

to

X

⁢

Y

over any ring

R

where

2

is invertible, with

M

=

(

1

-

1

1

1

)

.

In the case where

R

=

ℝ

is the

field of real

numbers (or any

formally real field

), we say that a quadratic form is

positive definite

,

negative definite

, or

positive semidefinite

according to whether its corresponding matrix is positive definite, negative definite, or positive semidefinite. The definiteness of a quadratic form is preserved under the equivalence relation on quadratic forms.

Sums of Quadratic Forms

If

Q

1

and

Q

2

are two quadratic forms in

m

and

n

indeterminates. We can define a quadratic form

Q

in

m

+

n

indeterminates in terms of

Q

1

and

Q

2

, called the

sum of

Q

1

and

Q

2

, as follows:

write

Q

1

=

𝐗

T

⁢

𝐀𝐗

and

Q

2

=

𝐘

T

⁢

𝐁𝐘

, with

𝐗

=

(

X

1

,

…

,

X

m

)

T

and

𝐘

=

(

Y

1

,

…

,

Y

n

)

T

. Then

Q

:=

𝐙

T

⁢

(

𝐀

⊕

𝐁

)

⁢

𝐙

,

where

𝐙

=

(

𝐗

,

𝐘

)

=

(

X

1

,

…

,

X

m

,

Y

1

,

…

,

Y

n

)

T

, and

𝐀

⊕

𝐁

is the

direct sum of matrices

𝐀

and

𝐁

.

Expressed in terms of

Q

1

and

Q

2

, we write

Q

=

Q

1

⊕

Q

2

. For example, if

Q

1

=

5

⁢

X

1

2

+

6

⁢

X

2

2

and

Q

2

=

10

⁢

X

1

⁢

X

2

, then

Q

1

⊕

Q

2

=

5

⁢

X

1

2

+

6

⁢

X

2

2

+

10

⁢

X

3

⁢

X

4

,

not

5

⁢

X

1

2

+

6

⁢

X

2

2

+

10

⁢

X

1

⁢

X

2

(

=

Q

1

+

Q

2

)

.

References

1

T. Y. Lam,

Introduction

to Quadratic Forms over Fields

, American Mathematical Society (2004)

Title

quadratic form

Canonical name

QuadraticForm

Date of creation

2013-03-22 12:19:22

Last modified on

2013-03-22 12:19:22

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

45

Author

rm50 (10146)

Entry type

Definition

Classification

msc 11E08

Classification

msc 11E04

Classification

msc 15A63

Related topic

PositiveDefinite

Related topic

NegativeDefinite

Related topic

SymmetricBilinearForm

Related topic

QuadraticSpace

Related topic

ProofOfGaussianMaximizesEntropyForGivenCovariance

Related topic

IsotropicQuadraticSpace

Defines

equivalent quadratic forms

Defines

sum of quadratic forms

Defines

evaluation of a quadratic form

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