decomposition of orthogonal operators as rotations and reflections

decomposition of orthogonal operators as rotations and reflections

Theorem 1

.

Let

V

be a

n

-dimensional real

inner product space

(

0

<

n

<

∞

). Then every

orthogonal

operator

T

on

V

can be decomposed into a series of two-dimensional

rotations

and one-dimensional

reflections

on mutually orthogonal subspaces of

V

.

We first explain the general idea behind the proof. Consider a rotation

R

of angle

θ

in a two-dimensional space. From its

orthonormal basis

representation

[

cos

⁡

θ

-

sin

⁡

θ

sin

⁡

θ

cos

⁡

θ

]

,

we find that the

characteristic polynomial

of

R

is

t

2

-

2

⁢

cos

⁡

θ

+

1

,
with (

complex

) roots

e

±

i

⁢

θ

.
(Not surprising, since

multiplication

by

e

i

⁢

θ

in the

complex plane

is rotation
by

θ

.)
Thus given the characteristic polynomial of

R

,
we can almost recover

1

1

Because the

complex roots

occur in

conjugate

pairs,
the information about the sign of

θ

is lost. This too, is not a surprise,
because the sign of

θ

, i.e. whether the rotation is clockwise or counterclockwise,
is dependent on the

orientation

of the basis vectors.

its rotation angle.

In the case of a reflection

S

, the

eigenvalues

of

S

are

-

1

and

1

; so again, the characteristic polynomial of

S

will provide some information on

S

.

So in

n

dimensions

, we are also going to look at the complex eigenvalues and eigenvectors of

T

to recover information about the rotations represented by

T

.

But there is one technical point —

T

is a

transformation

on a

real vector
space

, so it does not really have “complex eigenvalues and eigenvectors”.
To make this

concept

rigorous

, we must consider the

complexification

T

ℂ

of

T

,
the operator defined by

T

ℂ

⁢

(

x

+

i

⁢

y

)

=

T

⁢

x

+

i

⁢

T

⁢

y

in the

vector space

V

ℂ

consisting of elements of the form

x

+

i

⁢

y

, for

x

,

y

∈

V

.
(For more details, see the entry on complexification (

http://planetmath.org/ComplexificationOfVectorSpace

).)

Lemma 1

.

For any

linear operator

T

:

V

→

V

, there exists a one- or two-dimensional

subspace

W

which is

invariant

under

T

.

Proof.

Consider

T

ℂ

and its characteristic polynomial.
By the

Fundamental Theorem of Algebra

, the characteristic polynomial has a complex root

λ

=

α

+

i

⁢

β

. Then there is an

eigenvector

x

+

i

⁢

y

≠

0

with eigenvalue

λ

. We have

T

⁢

x

+

i

⁢

T

⁢

y

=

T

ℂ

⁢

(

x

+

i

⁢

y

)

=

λ

⁢

(

x

+

i

⁢

y

)

=

(

α

⁢

x

-

β

⁢

y

)

+

i

⁢

(

β

⁢

x

+

α

⁢

y

)

.

Equating real and

imaginary

components

,
we see that

T

⁢

x

=

α

⁢

x

-

β

⁢

y

∈

W

T

⁢

y

=

β

⁢

x

+

α

⁢

y

∈

W

where

W

=

span

⁡

{

x

,

y

}

.

W

is two-dimensional if

x

and

y

are

linearly independent

; otherwise it is one-dimensional

2

2

In fact, the space

W

constructed is two-dimensional if and only if the eigenvalue

λ

is not purely real. Compare with the remark (i) after
the proof of

Theorem

1.
Actually, Lemma 1 has more uses than just proving Theorem 1.
For example, if

x

˙

=

A

⁢

x

is a

linear differential equation

,
and the constant coefficient matrix

A

has only simple eigenvalues,
then it is a

consequence

of Lemma 1, that the

differential equation

decomposes into a series of

disjoint

one-variable and two-variable equations.
The solutions are then readily understood: they are always of the form of an
exponential multiplied by a sinusoid, and

linear combinations

thereof.
The sinusoids will be present whenever there are non-real eigenvalues.

. And we have

T

⁢

(

W

)

⊆

W

as claimed.
∎

Proof of Theorem 1.

We recursively factor

T

; formally, the proof will be by

induction

on the dimension

n

.

The case

n

=

1

is

trivial

. We have

det

⁡

T

=

±

1

; if

det

⁡

T

=

1

then

T

is the

identity

; otherwise

T

⁢

x

=

-

x

is a reflection.

For larger

n

, by Lemma 1 there exists a

T

-

invariant subspace

W

. The

orthogonal complement

W

⟂

of

W

is

T

-invariant also, because
for all

x

∈

W

⟂

and

y

∈

W

,

⟨

T

⁢

x

,

y

⟩

=

⟨

x

,

T

-

1

⁢

y

⟩

because

T

preserves

inner product

=

0

because

T

-

1

⁢

(

W

)

=

W

.

Let

T

W

be the operator that acts as

T

on

W

and is the identity on

W

⟂

.
Similarly, let

T

W

⟂

be the operator that acts as

T

on

W

⟂

and is the identity on

W

. Then

T

=

T

W

∘

T

W

⟂

.

T

W

restricted to

W

is orthogonal, and since

W

is one- or two-dimensional,

T

W

must therefore be a rotation or reflection (or the identity) in a line or plane.

T

W

⟂

restricted to

W

⟂

is also orthogonal.

W

⟂

has dimension

<

n

,
so by the

induction hypothesis

we can continue to factor it into operators acting on subspaces of

W

⟂

that are mutually orthogonal. These subspaces will of course also be orthogonal to

W

.

The proof is now

complete

, except that we did not rule out

T

W

being a reflection even
when

W

is two-dimensional.
But of course, if

T

W

is a reflection in two dimensions,
then it can be factored as a reflection on a one-dimensional subspace

span

⁡

{

v

}

composed with the identity on

span

{

v

}

⟂

.
∎

Actually, in the third paragraph of the proof, we implicitly assumed that every orthogonal operator on two-dimensional is either a rotation or reflection.
This is a well-known fact, but it does need to be formally proven:

Theorem 2

.

If

V

is a two-dimensional real inner product space, then every orthogonal operator

T

is either a rotation or reflection.

Proof.

Fix any orthonormal basis

{

e

1

,

e

2

}

for

V

.
Since

T

is orthogonal,

∥

T

⁢

e

1

∥

=

∥

e

1

∥

=

1

, i.e.

T

⁢

e

1

is a

unit vector

on the plane, so there exists an angle

θ

(unique modulo

2

⁢

π

)
such that

T

⁢

e

1

=

cos

⁡

θ

⁢

e

1

+

sin

⁡

θ

⁢

e

2

.
Similarly

T

⁢

e

2

is a unit vector, but since

e

1

and

e

2

are orthogonal, so
are

T

⁢

e

1

and

T

⁢

e

2

. Then it is found that the solution to

T

⁢

e

2

must be either

-

sin

⁡

θ

⁢

e

1

+

cos

⁡

θ

⁢

e

2

or

sin

⁡

θ

⁢

e

1

-

cos

⁡

θ

⁢

e

2

. Putting all this together, the matrix for

T

is:

[

cos

⁡

θ

∓

sin

⁡

θ

sin

⁡

θ

±

cos

⁡

θ

]

.

The first solution for

T

⁢

e

2

corresponds to a

rotation matrix

(and

det

⁡

T

=

1

);
the second solution for

T

⁢

e

2

corresponds to a
reflection matrix (

http://planetmath.org/DerivationOf2DReflectionMatrix

) (and

det

⁡

T

=

-

1

).
∎

1

Remarks

i.

Observe that the two equations for

T

⁢

x

and

T

⁢

y

appearing in the proof of Lemma 1
do specify a rotation of angle

±

θ

when

λ

=

α

+

i

⁢

β

=

e

i

⁢

θ

and

x

,

y

are orthonormal.
So by examining the complex eigenvales and eigenvectors of

T

ℂ

,
we can reconstruct the rotation.

This construction can be used to give an alternate proof of Theorem 1; we sketch it below:

The complexified space

V

ℂ

has an inner product

structure

inherited from

V

(again, see
complexification (

http://planetmath.org/ComplexificationOfVectorSpace

) for details).
Let

U

=

T

ℂ

. Since

T

is orthogonal,

U

is

unitary

, and hence normal (

U

*

⁢

U

=

U

⁢

U

*

).
There exists an orthonormal basis of eigenvectors for

U

(the

Schur decomposition

(

http://planetmath.org/CorollaryOfSchurDecomposition

)).
Let

z

=

x

+

i

⁢

y

is any one of these eigenvectors with a complex, non-real eigenvalue

λ

. Then

|

λ

|

=

1

because

U

is unitary,
and

z

¯

=

x

-

i

⁢

y

is another eigenvector with eigenvalue

λ

¯

.
Using the

V

ℂ

inner product

formula

, the vectors

x

/

2

and

y

/

2

can be shown to be orthonormal.
Then the proof of Lemma 1 shows that

T

acts as a rotation in the plane

span

⁡

{

x

,

y

}

.
All such planes obtained will be orthogonal to each other.

To summarize, the orthogonal subplanes of rotation are found by grouping conjugate pairs
of complex eigenvectors. If one actually needs to determine the planes of rotation explicitly (for dimensions

n

≥

4

), then probably it is better to work directly with the complexified matrix,
rather than to factor the matrix over the reals.

ii.

The decomposition of

T

is not unique. However, it is always possible to obtain
a decomposition which contains at most one reflection,
because any two single-dimensional reflections can always be combined into
a two-dimensional rotation.
In any case, the parity of the number of reflections in a decomposition of

T

is invariant,
because the parity is equal to

det

⁡

T

.

iii.

In the decomposition, the component rotations and reflections all commute
because they act on orthogonal subspaces.

iv.

If we take a basis for

V

describing the mutually orthogonal subspaces in Theorem 1,
the matrix for

T

looks like:

[

cos

⁡

θ

1

-

sin

⁡

θ

1

sin

⁡

θ

1

cos

⁡

θ

1

⋱

cos

⁡

θ

k

-

sin

⁡

θ

k

sin

⁡

θ

k

cos

⁡

θ

k

±

1

+

1

⋱

+

1

]

where

θ

1

,

…

,

θ

k

are the rotation angles, one for each orthogonal subplane,
and the

±

1

in the middle is the reflective component (if present). The rest of the entries
in the matrix are zero.

v.

Sometimes any orthogonal operator

T

with

det

⁡

T

=

1

is called a rotation, even though

strictly

speaking it is actually series of rotations
(each on different “axes”). Similarly, when

det

⁡

T

=

-

1

,

T

may be called a reflection, even though again it is not always a single (one-dimensional)
reflection.

In this

language

, a rotation composed with a rotation will always be a rotation;
a rotation composed with a reflection is a reflection; and two reflections composed together will always be a rotation.

vi.

In

ℝ

3

, an orthogonal operator with

positive

determinant

is necessarily
a rotation on one axis which is left fixed (except when the operator is the identity).
This follows simply because there is no way to fit more than one orthogonal subplane
into three-dimensional space.

A

composition

of two rotations in

ℝ

3

would then
be a rotation too.
On the other hand it is not at all

obvious

what

relation

the axis of rotation
of the composition has with the original two axes of rotation.

For an

explicit formula

for a rotation matrix in

ℝ

3

that does not
require manual calculation of the basis vectors for the rotation subplane, see Rodrigues’

rotation formula

.

vii.

In

ℝ

n

, reflections can be carried out by first

embedding

ℝ

n

into

ℝ

n

+

1

and then rotating

ℝ

n

+

1

.
(Here, the words “rotation” and “reflection” being taken in their extended sense of (v).)
For example, in the plane, a right hand can be
rotated in

ℝ

3

into a left hand.

To be specific, suppose we embed

ℝ

n

in

ℝ

n

+

1

as the first

n

coordinates.
Then we gain an extra degree of freedom in the last coordinate of

ℝ

n

+

1

(with coordinate vector

e

n

+

1

).
Given an orthogonal operator

T

:

ℝ

n

→

ℝ

n

,
we can extend it to an operator on

T

′

:

ℝ

n

+

1

→

ℝ

n

+

1

by having it act as

T

on the lower

n

coordinates,
and setting

T

′

⁢

(

e

n

+

1

)

=

-

e

n

+

1

.
Since

det

⁡

T

′

=

-

det

⁡

T

=

1

,
our new

T

′

will be a rotation (the extra angle of rotation will be by

π

)
that reflects sets in the

ℝ

n

plane.

References

1

Friedberg, Insel, Spence.

Linear Algebra

. Prentice-Hall, 1997.

2

Vladimir I. Arnol’d (trans. Roger Cooke).

Ordinary Differential Equations

. Springer-Verlag, 1992.

Title

decomposition of orthogonal operators as rotations and reflections

Canonical

name

DecompositionOfOrthogonalOperatorsAsRotationsAndReflections

Date of creation

2013-03-22 15:24:11

Last modified on

2013-03-22 15:24:11

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

15

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 15A04

Related topic

RotationMatrix

Related topic

OrthogonalMatrices

Related topic

DimensionOfTheSpecialOrthogonalGroup

Related topic

RodriguesRotationFormula

Related topic

DerivationOfRotationMatrixUsingPolarCoordinates

Related topic

DerivationOf2DReflectionMatrix

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