Cayley’s parameterization of orthogonal matrices

Cayley’s parameterization of orthogonal matrices

Any

orthogonal matrix

O

which does not have

-

1

as an

eigenvalue

can be expressed as

O

=

(

I

+

A

)

⁢

(

I

-

A

)

-

1

for some suitable

skew-symmetric matrix

A

.

Conversely

, any skew-symmetric matrix

A

can be expressed in terms of a suitable orthogonal matrix

O

by a

similar

formula

,

A

=

(

O

+

I

)

-

1

⁢

(

O

-

I

)

.

These two formulae are each other’s

inverses

and set up a

one-to-one correspondence

between

orthogonal

and skew-symmetric matrices.

0.0.1

Proof

The

restriction

on the eigenvalues of

O

is necessary in order for

I

+

O

to be

invertible

.

It is a matter of simple computation why these formulae are correct. Suppose that

A

is

skew-symmetric

. Then

O

T

⁢

O

=

(

(

I

+

A

)

⁢

(

I

-

A

)

-

1

)

T

⁢

(

I

+

A

)

⁢

(

I

-

A

)

-

1

Using the fact that the

transpose

of a

product

is the product of the transposes in the opposite order,

=

(

(

I

-

A

)

-

1

)

T

⁢

(

I

+

A

)

T

⁢

(

I

+

A

)

⁢

(

I

-

A

)

-

1

Using the fact that the transpose of a sum is the sum of transposes and the transpose of an inverse is the inverse of the transpose,

=

(

I

T

-

A

T

)

-

1

⁢

(

I

T

+

A

T

)

⁢

(

I

+

A

)

⁢

(

I

-

A

)

-

1

By the definition of skew-symmetry,

A

T

=

-

A

and

I

T

=

I

,

=

(

I

+

A

)

-

1

⁢

(

I

-

A

)

⁢

(

I

+

A

)

⁢

(

I

-

A

)

-

1

Finally, since

I

+

A

and

I

-

A

commute, we may switch the order of the second and third factors:

=

(

I

+

A

)

-

1

⁢

(

I

+

A

)

⁢

(

I

-

A

)

⁢

(

I

-

A

)

-

1

Then the first two factors and the last two factors cancel, showing that

O

T

⁢

O

=

I

.

Next, we verify that the second formula is indeed the inverse of the first formula. Multiplying by

I

-

A

on both sides,

O

⁢

(

I

-

A

)

=

I

+

A

Expanding this and moving terms from one side of the

equation

to the other,

O

-

I

=

A

+

O

⁢

A

Factoring,

O

-

I

=

(

I

+

O

)

⁢

A

Multiplying both sides by

(

I

+

O

)

-

1

, we obtain the desired formula:

(

O

+

I

)

-

1

⁢

(

O

-

I

)

=

A

Finally, one can show that, if

O

is orthogonal, then

A

is skew-symmetric using the same sort of computation that was used to show the

converse

:

A

T

=

(

(

O

+

I

)

-

1

⁢

(

O

-

I

)

)

T

Using the facts about transposes of sums, products, and inverses,

=

(

O

T

-

I

T

)

⁢

(

O

T

+

I

T

)

-

1

Since

O

is orthogonal,

O

T

=

O

-

1

. As usual

I

T

=

I

.

=

(

O

-

1

-

I

)

⁢

(

O

-

1

+

I

)

-

1

Insert an

identity matrix

between the two factors like so:

=

(

O

-

1

-

I

)

⁢

I

⁢

(

O

-

1

+

I

)

-

1

Replace the identity matrix with

O

⁢

O

-

1

:

=

(

O

-

1

-

I

)

⁢

O

⁢

O

-

1

⁢

(

O

-

1

+

I

)

-

1

Absorbing

the

O

and the

O

-

1

into the factors,

=

(

I

-

O

)

⁢

(

I

+

O

)

-

1

=

-

(

O

-

I

)

⁢

(

O

+

I

)

-

1

.

Since

O

⁢

(

O

+

I

)

-

1

=

(

(

O

+

I

)

⁢

O

T

)

-

1

=

(

I

+

O

T

)

-

1

=

(

O

T

⁢

(

O

+

I

)

)

-

1

=

(

O

+

I

)

-

1

⁢

O

,

(

O

-

I

)

and

(

O

+

I

)

-

1

commute, and consequently

=

-

(

O

+

I

)

-

1

⁢

(

O

-

I

)

=

-

A

.

0.0.2

Cayley Transform

The

relation

between

A

and

O

which is set up by the formulas

O

=

(

I

+

A

)

⁢

(

I

-

A

)

-

1

and

A

=

(

O

+

I

)

-

1

⁢

(

O

-

I

)

is sometimes known as the

Cayley transform

. Note that the proof that these two formulas are each other’s inverses did not require

A

to be skew-symmetric or

O

to be orthogonal. Hence, the Cayley transform is defined for all matrices such that

-

1

is not an eigenvalue of

O

. (Recall that this condition is necessary to insure that

O

+

I

is invertible.

0.0.3

Generalizations

The Cayley parameterization can be generalized to

unitary

transforms. Namely, if

U

is a

unitary matrix

, then

U

is the Cayley transform of a skew-Hermitean matrix

A

. Since a skew-Hermitean matrix can be written as

i

times a

Hermitean

matrix, the Cayley transform is often written as follows when dealing with unitary matrices:

U

=

(

i

⁢

I

+

H

)

⁢

(

i

⁢

I

-

H

)

-

1

i

⁢

H

=

(

U

+

I

)

-

1

⁢

(

U

-

I

)

where

H

is Hermitean. The proof in this case is substantially the same as was presented above; all one has to do is replace matrix

transposition

with Hermitean

conjugation

.

A special case of this worth pointing out is the case of one-dimensional unitary matrices. The sole entry of a one dimensional unitary matrix must have modulus 1 and the sole entry of a one-dimensional Hermitean matrix must be real. In that case, the Cayley transform reduces to

u

=

i

+

h

i

-

h

i

⁢

h

=

u

+

i

u

-

i

,

which is a fractional linear transform that maps the unit circle to the real axis.

The Cayley parameterization can be generalized to the case of a general

inner product

with arbitrary

signature

(see Sylvester’s law for the definition of signature — Cayley and Sylvester were the best of friends). We simply need to define the transpose of a matrix

M

by the condition

(

M

T

⁢

u

)

⋅

v

=

u

⋅

(

M

⁢

v

)

for all vectors

u

and

v

. In particular, this allows one to parameterize pseudo-orthogonal matrices such as Lorentz transformations using a Cayley parameterization. Likewise, given a

conjugate

linear inner product on a

complex vector space

, one has a Cayley parameterization of the unitary (or pseudo-unitary) transforms which preserve the product.

In

conclusion

, it might be worth pointing out that the Cayley transform generalizes to the case of

infinite

dimensions

, if one replaces matrices with

operators

on a

Hilbert space

. In particular, it is useful because unitary and orthogonal operators are

bounded

whereas Hermitean and skew-symmetric operators may or may not be bounded. For

instance

, it is often easier to obtain the spectral decomposition of a Hermitean operator or study

symmetric

extensions

of a

symmetric operator

by first performing a Cayley transform and dealing with the resulting

bounded operator

.

Title

Cayley’s parameterization of orthogonal matrices

Canonical name

CayleysParameterizationOfOrthogonalMatrices

Date of creation

2013-03-22 14:51:38

Last modified on

2013-03-22 14:51:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

20

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 22E70

Classification

msc 15A57

Defines

Cayley transform

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