Gram matrix

Gram matrix

For a

vector space

V

of

dimension

n

over a field

k

, endowed with an

inner product

<

.

,

.

>

:

V

×

V

→

k

, and for any given

sequence

of

elements

x

1

,

…

,

x

l

∈

V

, consider the following

inclusion map

ι

associated to the

(

x

i

)

i

=

1

,

…

,

l

:

ι

:

k

l

→

k

n

=

V

(

λ

1

,

…

,

λ

l

)

↦

∑

i

=

1

l

λ

i

⁢

x

i

The

Gram bilinear form

of the

(

x

i

)

i

=

1

,

…

,

l

is the

function

x

,

y

∈

k

l

×

k

l

↦

<

ι

(

x

)

,

ι

(

y

)

>

The

Gram matrix

of the

(

x

i

)

i

=

1

,

…

,

l

is the matrix associated to the Gram bilinear form in the

canonical basis

of

k

l

. The Gram form (resp. matrix) is a

symmetric bilinear form

(resp. matrix).

Gram forms/matrices are usually considered with

k

=

ℝ

and

<

.

,

.

>

the usual

scalar products

on vector spaces over

ℝ

. In that

context

they have a strong geometrical meaning:

•

The

determinant

of the Gram form/matrix is

0

iff

ι

is an

injection

.

•

If

ι

is injective, the Gram matrix (resp. form) is a

positive

symmetric matrix

(resp.

bilinear form

).

•

det

⁡

(

G

)

-

2

=

Vol

⁢

(

ι

-

1

⁢

(

B

0

,

1

)

)

where

G

is the gram form/matrix,

Vol

denotes the volume of a subset of

ℝ

n

, and

B

0

,

1

is the unit ball of

ℝ

n

centered at

0

.

•

Let

Δ

:

λ

∈

ℝ

l

↦

G

⁢

(

λ

,

λ

)

, where

G

is the Gram bilinear form, then

ι

-

1

⁢

(

B

0

,

1

)

=

Δ

-

1

⁢

(

[

0

,

1

]

)

.

•

If

ι

is injective and

s

:

Im

⁢

(

ι

)

⊂

ℝ

n

→

ℝ

l

is an

isometry

, then

det

⁡

(

s

∘

ι

)

2

=

det

⁡

(

G

)

.

•

Let

f

be an

endomorphism

of

ℝ

n

and

M

its matrix. Let

H

,

U

be the

polar decomposition

of

M

,

H

is a

symmetric

positive matrix

and

U

an

orthogonal matrix

. Let the

x

i

be the columns of

M

(

l

=

n

) and let

G

be the Gram matrix the

x

i

. Then

H

2

=

G

. (N.B.: this is one way to prove the

existence

of the polar decomposition, take the

square root

of the Gram matrix, multiply

M

by its

inverse

and it easily follows that what is obtained is an orthogonal matrix).

They are utilized in statistics in

Principal components analysis

. One wants to determine the general trend in terms of few

characteristics

(

l

of them) in a large sample (

n

individuals

). Each

x

i

(

∈

ℝ

n

)

represents the results of the

n

individuals in the sample for the

i

th

characteristic. Each one of the

l

dimensions represents a characteristic and one wants to know what are the predominant characteristics and if they bear some kind of linear

relations

between them. This is achieved by diagonalizing the Gram matrix (often called dispersion matrix or

covariance matrix

in that context). The higher the

eigenvalue

, the more important the

eigenvector

associated to it.

Title

Gram matrix

Canonical

name

GramMatrix

Date of creation

2013-03-22 17:56:55

Last modified on

2013-03-22 17:56:55

Owner

lalberti (18937)

Last modified by

lalberti (18937)

Numerical id

5

Author

lalberti (18937)

Entry type

Definition

Classification

msc 15A63

Synonym

Gram matrices

Synonym

Gram bilinear form

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