bilinear form

bilinear form

Definition.

Let

U

,

V

,

W

be

vector spaces

over a field

K

. A

bilinear map

is a

function

B

:

U

×

V

→

W

such that

1.

the map

x

↦

B

⁢

(

x

,

y

)

from

U

to

W

is linear for each

y

∈

V

2.

the map

y

↦

B

⁢

(

x

,

y

)

from

V

to

W

is linear for each

x

∈

U

.

That is,

B

is

bilinear

if it is linear in each parameter,
taken separately.

Bilinear forms.

A

bilinear form

is a bilinear map

B

:

V

×

V

→

K

. A

W

-valued bilinear form is a bilinear map

B

:

V

×

V

→

W

. One
often encounters bilinear forms with additional

assumptions

. A
bilinear form is called

•

symmetric

if

B

⁢

(

x

,

y

)

=

B

⁢

(

y

,

x

)

,

x

,

y

∈

V

;

•

skew-symmetric

if

B

⁢

(

x

,

y

)

=

-

B

⁢

(

y

,

x

)

,

x

,

y

∈

V

;

•

alternating

if

B

⁢

(

x

,

x

)

=

0

,

x

∈

V

.

By expanding

B

⁢

(

x

+

y

,

x

+

y

)

=

0

, we can show alternating implies
skew-symmetric. Further if

K

is not of

characteristic

2

, then
skew-symmetric implies alternating.

Left and Right Maps.

Let

B

:

U

×

V

→

W

be a bilinear map. We may identify

B

with the

linear map

B

⊗

:

U

⊗

V

→

W

(see

tensor product

). We may
also identify

B

with the linear maps

B

L

:

U

→

L

⁢

(

V

,

W

)

,

B

L

⁢

(

x

)

⁢

(

y

)

=

B

⁢

(

x

,

y

)

,

x

∈

U

,

y

∈

V

;

B

R

:

V

→

L

⁢

(

U

,

W

)

,

B

R

⁢

(

y

)

⁢

(

x

)

=

B

⁢

(

x

,

y

)

,

x

∈

U

,

y

∈

V

.

called the left and right map, respectively.

Next, suppose that

B

:

V

×

V

→

K

is a bilinear form. Then both

B

L

and

B

R

are linear maps from

V

to

V

*

, the

dual vector
space

of

V

. We can therefore say that

B

is symmetric if and only
if

B

L

=

B

R

and that

B

is anti-symmetric if and only if

B

L

=

-

B

R

. If

V

is

finite-dimensional

, we can identify

V

and

V

*

*

, and assert that

B

L

=

(

B

R

)

*

; the left and right maps are, in
fact,

dual homomorphisms

.

Rank.

Let

B

:

U

×

V

→

K

be a bilinear form, and suppose that

U

,

V

are
finite dimensional. One can show that

rank

⁡

B

L

=

rank

⁡

B

R

. We call
this integer

rank

⁡

B

, the

of

B

.
Applying the

rank-nullity theorem

to both the left and right maps
gives the following results:

dim

⁡

U

=

dim

⁡

ker

⁡

B

L

+

rank

⁡

B

dim

⁡

V

=

dim

⁡

ker

⁡

B

R

+

rank

⁡

B

We say that

B

is

non-degenerate

if both the left and right map are
non-degenerate. Note that in for

B

to be non-degenerate it is
necessary that

dim

⁡

U

=

dim

⁡

V

. If this holds, then

B

is
non-degenerate if and only if

rank

⁡

B

is equal to

dim

⁡

U

,

dim

⁡

V

.

Orthogonal complements.

Let

B

:

V

×

V

→

K

be a bilinear form, and let

S

⊂

V

be a

subspace

. The left and right

orthogonal complements

of

S

are
subspaces

S

⟂

,

S

⟂

⊂

V

defined as follows:

S

⟂

=

{

u

∈

V

∣

B

⁢

(

u

,

v

)

=

0

⁢

for all

⁢

v

∈

S

}

,

S

⟂

=

{

v

∈

V

∣

B

⁢

(

u

,

v

)

=

0

⁢

for all

⁢

u

∈

S

}

.

We may also realize

S

⟂

by considering the linear map

B

R

′

:

V

→

S

*

obtained as the composition of

B

R

:

V

→

V

*

and the dual homomorphism

V

*

→

S

*

.
Indeed,

S

⟂

=

ker

⁡

B

R

′

. An analogous
statement can be made for

S

⟂

.

Next, suppose that

B

is non-degenerate.
By the rank-nullity
theorem we have that

dim

⁡

V

=

dim

⁡

S

+

dim

⁡

S

⟂

=

dim

⁡

S

+

dim

⁡

S

⟂

.

Therefore, if

B

is non-degenerate, then

dim

⁡

S

⟂

=

dim

⁡

S

⟂

.

Indeed, more can be said if

B

is either symmetric or
skew-symmetric. In this case, we actually have

S

⟂

=

S

⟂

.

We say that

S

⊂

V

is a non-degenerate subspace relative to

B

if the

restriction

of

B

to

S

×

S

is non-degenerate. Thus,

S

is a non-degenerate subspace if and only if

S

∩

S

⟂

=

{

0

}

,
and also

S

∩

S

⟂

=

{

0

}

. Hence, if

B

is non-degenerate
and if

S

is a non-degenerate
subspace, we have

V

=

S

⊕

S

⟂

=

S

⊕

S

⟂

.

Finally, note that if

B

is positive-definite, then

B

is necessarily non-degenerate and
that every subspace is non-degenerate. In this way we arrive at the
following well-known result: if

V

is positive-definite

inner product
space

, then

V

=

S

⊕

S

⟂

for every subspace

S

⊂

V

.

Adjoints.

Let

B

:

V

×

V

→

K

be a non-degenerate bilinear
form, and let

T

∈

L

⁢

(

V

,

V

)

be a linear

endomorphism

. We define the

right adjoint

T

⋆

∈

L

⁢

(

V

,

V

)

to be the unique linear map such that

B

⁢

(

T

⁢

u

,

v

)

=

B

⁢

(

u

,

T

⋆

⁢

v

)

,

u

,

v

∈

V

.

Letting

T

∗

:

V

∗

→

V

∗

denote the dual homomorphism,
we also have

T

⋆

=

B

R

-

1

∘

T

∗

∘

B

R

.

Similarly, we define the left adjoint

T

⋆

∈

L

⁢

(

V

,

V

)

by

T

⋆

=

B

L

-

1

∘

T

∗

∘

B

L

.

We then have

B

⁢

(

u

,

T

⁢

v

)

=

B

⁢

(

T

⋆

⁢

u

,

v

)

,

u

,

v

∈

V

.

If

B

is either symmetric or skew-symmetric, then

T

⋆

=

T

⋆

, and we simply use

T

⋆

to refer to the adjoint homomorphism.

Additional remarks.

1.

if

B

is a symmetric, non-degenerate bilinear form,
then the

adjoint

operation

is represented, relative to an

orthogonal
basis

(if one exists), by the matrix transpose.

2.

If

B

is a symmetric, non-degenerate bilinear form
then

T

∈

L

⁢

(

V

,

V

)

is then said to be a

normal operator

(with
respect to

B

) if

T

commutes with its adjoint

T

⋆

.

3.

An

n

×

m

matrix may be regarded as a bilinear form over

K

n

×

K

m

. Two such matrices,

B

and

C

, are said to be

congruent

if there exists an

invertible

P

such that

B

=

P

T

⁢

C

⁢

P

.

4.

The

identity matrix

,

I

n

on

ℝ

n

×

ℝ

n

gives the standard

Euclidean

inner product

on

ℝ

n

.

Title

bilinear form

Canonical name

BilinearForm

Date of creation

2013-03-22 12:14:02

Last modified on

2013-03-22 12:14:02

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

54

Author

rmilson (146)

Entry type

Definition

Classification

msc 47A07

Classification

msc 11E39

Classification

msc 15A63

Synonym

bilinear

Related topic

DualityWithRespectToANonDegenerateBilinearForm

Related topic

BilinearMap

Related topic

Multilinear

Related topic

SkewSymmetricBilinearForm

Related topic

SymmetricBilinearForm

Related topic

NonDegenerateBilinearForm

Defines

rank of bilinear form

Defines

left map

Defines

right map

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