7Transformation groups

IA Vectors and Matrices

7.1 Groups of orthogonal matrices

Proposition.

The set of all

n × n

orthogonal matrices

P

forms a group under

matrix multiplication.

Proof.

0.

If

P, Q

are orthogonal, then consider

R

=

P Q

.

RR

T

= (

P Q

)(

P Q

)

T

=

P (QQ

T

)P

T

= P P

T

= I. So R is orthogonal.

1. I satisfies II

T

= I. So I is orthogonal and is an identity of the group.

2.

Inverse: if

P

is orthogonal, then

P

−1

=

P

T

by definition, which is also

orthogonal.

3.

Matrix multiplication is associative since function composition is associative.

Definition

(Orthogonal group)

.

The orthogonal group

O

(

n

) is the group of

orthogonal matrices.

Definition

(Special orthogonal group)

.

The special orthogonal group is the

subgroup of O(n) that consists of all orthogonal matrices with determinant 1.

In general, we can show that any matrix in O(2) is of the form

cos θ −sin θ

sin θ cos θ

or

cos θ sin θ

sin θ −cos θ