Gram determinant

Gram determinant

Let

V

be an

inner product space

over a field

k

with

⟨

⋅

,

⋅

⟩

the

inner product

on

V

(note: since

k

is not restricted to be either

ℝ

or

ℂ

, the inner product here shall mean a

symmetric bilinear form

on

V

). Let

x

1

,

x

2

,

…

,

x

n

be arbitrary vectors in

V

. Set

r

i

⁢

j

=

⟨

x

i

,

x

j

⟩

. The

Gram determinant

of

x

1

,

x

2

,

…

,

x

n

is defined to be the

determinant

of the

symmetric matrix

(

r

11

⋯

r

1

⁢

n

⋮

⋱

⋮

r

n

⁢

1

⋯

r

n

⁢

n

)

Let’s denote this determinant by

Gram

⁡

[

x

1

,

x

2

,

…

,

x

n

]

.

Properties

.

1.

Gram

⁡

[

x

1

,

…

,

x

i

,

…

,

x

j

,

…

,

x

n

]

=

Gram

⁡

[

x

1

,

…

,

x

j

,

…

,

x

i

,

…

,

x

n

]

. More generally,

Gram

⁡

[

x

1

,

…

,

x

n

]

=

Gram

⁡

[

x

σ

⁢

(

1

)

,

…

,

x

σ

⁢

(

n

)

]

, where

σ

is a

permutation

on

{

1

,

…

,

n

}

.

2.

Gram

⁡

[

x

1

,

…

,

a

⁢

x

i

+

b

⁢

x

j

,

…

,

x

j

,

…

,

x

n

]

=

a

2

⁢

Gram

⁡

[

x

1

,

…

,

x

i

,

…

,

x

j

,

…

,

x

n

]

,

a

,

b

∈

k

.

3.

Setting

a

=

0

and

b

=

1

in Property 2, we get

Gram

⁡

[

x

1

,

…

,

x

j

,

…

,

x

j

,

…

,

x

n

]

=

0

.

4.

Properties 2 and 3 can be generalized as follows: if

x

i

(in the

i

th term) is replaced by a

linear combination

y

=

r

1

⁢

x

1

+

⋯

+

r

n

⁢

x

n

, then

Gram

⁡

[

x

1

,

…

,

y

,

…

,

x

n

]

=

r

i

2

⁢

Gram

⁡

[

x

1

,

…

,

x

i

,

…

,

x

n

]

.

5.

Suppose

k

is an

ordered field

. Then it can be shown that the Gram determinant is at least 0, and at most the product

⟨

x

1

,

x

1

⟩

⁢

⋯

⁢

⟨

x

n

,

x

n

⟩

.

6.

Suppose that in

addition

to

k

being ordered, that every

positive element

in

k

is a square, then the Gram determinant is equal to the square of the volume of the (hyper)

parallelepiped

generated by

x

1

,

…

,

x

n

. (Recall that an

n

-dimensional parallelepiped is the set of vectors which are linear combinations of the form

r

1

⁢

x

1

+

…

+

r

n

⁢

x

n

where

0

≤

r

i

≤

1

.)

7.

It’s now easy to see that in Property 5, the Gram determinant is 0 if the

x

i

’s are

linearly dependent

, and attains its maximum if the

x

i

’s are pairwise

orthogonal

(a quick proof: in the above matrix,

r

i

⁢

j

=

0

if

i

≠

j

), which corresponds exactly to the square of the volume of the hyperparallelepiped

spanned by

the

x

i

’s.

8.

If

e

1

,

…

,

e

n

are basis elements of a

quadratic space

V

over an order field whose positive elements are squares, then

V

is , or , iff

Gram

⁡

[

e

1

,

…

,

e

n

]

=

0

.

References

1

Georgi E. Shilov, “An Introduction to the Theory of

Linear Spaces

”, translated from Russian by Richard A. Silverman, 2nd Printing, Prentice-Hall, 1963.

Title

Gram determinant

Canonical

name

GramDeterminant

Date of creation

2013-03-22 15:41:37

Last modified on

2013-03-22 15:41:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Related topic

GrammianDeterminant

Related topic

GramMatrix

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