8Matrix groups

IA Groups

8.2 Actions of GL

n

(C)

Proposition. GL

n

(

C

) acts faithfully on

C

n

by left multiplication to the vector,

with two orbits (0 and everything else).

Proof. First show that it is a group action:

1. If A ∈ GL

n

(C) and v ∈ C

n

, then Av ∈ C

n

. So it is closed.

2. Iv = v for all v ∈ C

n

.

3. A(Bv) = (AB)v.

Now prove that it is faithful: a linear map is determined by what it does

on a basis. Take the standard basis

e

1

= (1

,

0

, · · · ,

0)

, · · · e

n

= (0

, · · · ,

1). Any

matrix which maps each

e

k

to itself must be

I

(since the columns of a matrix

are the images of the basis vectors)

To show that there are 2 orbits, we know that

A0

=

0

for all

A

. Also, as

A

is invertible,

Av

=

0 ⇔ v

=

0

. So

0

forms a singleton orbit. Then given any

two vectors

v 6

=

w ∈ C

n

\ {

0

}

, there is a matrix

A ∈ GL

n

(

C

) such that

Av

=

w

(cf. Vectors and Matrices).

Similarly, GL

n

(R) acts on R

n

.

Proposition. GL

n

(C) acts on M

n×n

(C) by conjugation. (Proof is trivial)

This action can be thought of as a “change of basis” action. Two matrices

are conjugate if they represent the same map but with respect to different bases.

The P is the base change matrix.

From Vectors and Matrices, we know that there are three different types of

orbits for GL

2

(C): A is conjugate to a matrix of one of these forms:

(i)

λ 0

0 µ

, with λ 6= µ, i.e. two distinct eigenvalues

(ii)

λ 0

0 λ

, i.e. a repeated eigenvalue with 2-dimensional eigenspace

(iii)

λ 1

0 λ

, i.e. a repeated eigenvalue with a 1-dimensional eigenspace

Note that we said there are three types of orbits, not three orbits. There are

infinitely many orbits, e.g. one for each of λI.