dimension of the special orthogonal group

dimension of the special orthogonal group

Let

V

be a

n

-dimensional real

inner product space

.
The group of

orthogonal

operators

on

V

with

positive

determinant

(i.e. the group of “

rotations

” on

V

)
is called the

special orthogonal group

, denoted

S

⁢

O

⁢

(

n

)

.

The

theorem

on decomposing orthogonal operators as rotations and

reflections

(

http://

planetmath

.org/DecompositionOfOrthogonalOperatorsAsRotationsAndReflections

)
suggests that all elements of

S

⁢

O

⁢

(

n

)

are all fundamentally two-dimensional in some sense.
This article is an

elementary

exploration of this aspect.

First observe that the set of orthogonal operators

O

⁢

(

n

)

is a manifold embedded
in the

real vector space

GL

⁢

(

V

)

≃

ℝ

n

×

n

,
defined by
the condition

f

⁢

(

X

)

=

X

⁢

X

*

-

I

=

0

, where

X

∈

G

⁢

L

⁢

(

V

)

, and

X

*

is the

transpose

of

X

(represented as a matrix in orthonormal

coordinates

).

f

⁢

(

X

)

=

0

is simply shorthand for the condition that the columns of

X

are
orthonormal — we see there are

n

+

n

⁢

(

n

-

1

)

/

2

=

n

⁢

(

n

+

1

)

/

2

scalar

equations

1

1

f

⁢

(

X

)

=

0

viewed as a

matrix equation

of course has

n

2

scalar equations,
but many of them are duplicated. For simplicity,
we prefer to view

f

’s

codomain

to be the space of

symmetric matrices

, which is

n

⁢

(

n

+

1

)

/

2

-dimensional.

in

n

2

variables. Then if we can show that the

derivative

D

⁡

f

⁢

(

X

)

as full rank everywhere,
then we will have established that

O

⁢

(

n

)

is a manifold of

dimension

n

⁢

(

n

-

1

)

/

2

.
This is an easy computation if organized in the right way:
for any

y

∈

GL

⁢

(

V

)

, by the

product rule

we have

D

⁡

f

⁢

(

X

)

⋅

Y

=

X

⁢

Y

*

+

Y

⁢

X

*

.

Consider first the case that

X

=

I

. Then

D

⁡

f

⁢

(

I

)

⋅

Y

=

0

implies

Y

*

=

-

Y

;
in other words,

Y

is

skew-symmetric

. But the

skew-symmetric matrices

have dimension

(

n

2

-

n

)

/

2

, so the kernel of

D

⁡

f

⁢

(

I

)

has this dimension.
By the dimension theorem of

linear algebra

, the rank of

D

⁡

f

⁢

(

I

)

is

n

2

-

(

n

2

-

n

)

/

2

=

n

⁢

(

n

+

1

)

/

2

,
which is the full rank.

For general

X

∈

O

⁢

(

n

)

, let

Y

′

=

X

*

⁢

Y

. Then

D

⁡

f

⁢

(

X

)

⋅

Y

=

0

reduces to

Y

′

⁣

*

=

-

Y

,
again the skew-symmetric condition.
The set of all such

Y

′

have dimension

(

n

2

-

n

)

/

2

; but

multiplication

by

X

*

is an

automorphism

of

GL

⁢

(

V

)

, so the set of all

Y

also have the same dimension.
Thus

D

⁡

f

⁢

(

X

)

has full rank, as before.

S

⁢

O

⁢

(

n

)

is of course just the part of

O

⁢

(

n

)

that

satisfies

the additional condition

det

⁡

X

=

1

. It has the same dimension as a manifold.

Notice that

n

⁢

(

n

-

1

)

/

2

=

(

n

2

)

, the number of ways to choose

2

basis elements out of

n

.
We might naïvely think that an

element

X

of

S

⁢

O

⁢

(

n

)

can be decomposed
into rotations on each of the coordinate 2-dimensional planes in

n

-dimensional space.
This does not quite work, because rotations do not commute, even though
we expect that the order of the

decomposition

should not matter (after all,
there is no canonical order on the pairs of basis elements of

V

).

Nevertheless, on an “

infinitesimal

level”, the decomposition does work.
Suppose we are given a one-parameter subgroup

2

2

Given

X

∈

S

⁢

O

⁢

(

n

)

, it can be shown with the orthogonal operator decomposition theorem that there is a one-parameter subgroup containing X. Hint:

e

A

=

(

cos

⁡

θ

-

sin

⁡

θ

sin

⁡

θ

cos

⁡

θ

)

,

if

⁢

A

=

(

0

-

θ

θ

0

)

.

X

t

of

S

⁢

O

⁢

(

n

)

.
From ODE theory,
it is known that

X

t

is given by the

matrix exponential

e

t

⁢

A

,
where

A

is derivative of

X

t

at

t

=

0

.
Also,

A

must be skew-symmetric, because it is a

tangent vector

to the curve

X

t

and so must lie in the

tangent space

to

S

⁢

O

⁢

(

n

)

(the kernel of

D

⁡

f

⁢

(

X

)

).

Since

X

t

=

e

t

⁢

A

, we may reasonably ask: in what way does the matrix

A

represent

the rotations

X

t

? The answer is suggested by the simpler case of

n

=

3

;
by Rodrigues’

rotation formula

,

A

is the operator

A

⁢

v

=

ω

×

v

where

ω

is the angular velocity vector of the rotation.

The

wedge product

of the exterior algebra on

V

generalizes the

cross product

in higher dimensions (

n

>

3

),
so we should start looking there,
and one cannot but help notice that

Λ

2

⁢

(

V

)

also has dimension

(

n

2

)

.
The basis elements of this space are of the form

x

^

∧

y

^

,
where

x

and

y

are

orthonormal basis

vectors of

V

, and

x

^

,

y

^

∈

V

*

are defined by

x

^

⁢

(

w

)

=

⟨

x

,

w

⟩

,

y

^

⁢

(

w

)

=

⟨

y

,

w

⟩

. By attempting to generalize the cross product

representation

for

A

, we can derive that the

natural isomorphism

between elements

η

∈

Λ

2

⁢

(

V

)

and the skew-symmetric matrices

A

should be given by:

⟨

A

⁢

v

,

w

⟩

=

η

⁢

(

v

,

w

)

.

If

η

=

ω

⁢

x

^

∧

y

^

,
a simple calculation shows

X

t

=

e

t

⁢

A

is a rotation on the plane spanned by

x

and

y

with angular velocity

ω

!

So one can think of the infinitesimal decomposition of

X

t

, as the “angular velocity” with its

(

n

2

)

components

. The fact that angular velocity in our physical world
can be represented by one vector, and the fact that rotations in

ℝ

3

have a single axis,
is a

consequence

of the fact that

(

3

2

)

=

3

.

So although rotations do not commute in general, the correspondence between rotations
and

alternating

2-tensors, shows that rotations do add up
and commute at the infinitesimal level

3

3

This really just boils down to the fact that a manifold looks like a

vector space

locally.

.

Title

dimension of the special orthogonal group

Canonical name

DimensionOfTheSpecialOrthogonalGroup

Date of creation

2013-03-22 15:24:22

Last modified on

2013-03-22 15:24:22

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Result

Classification

msc 15A04

Classification

msc 15A75

Classification

msc 20G20

Related topic

ProofOfRodriguesRotationFormula

Related topic

DecompositionOfOrthogonalOperatorsAsRotationsAndReflections

Related topic

OrthogonalMatrices

Related topic

ExteriorAlgebra

Related topic

OrthogonalGroup

Defines

special orthogonal group

Generated on Fri Feb 9 18:21:18 2018 by

LaTeXML