8.2 Actions of GL
) acts faithfully on
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
(C) and v ∈ C
, then Av ∈ C
. So it is closed.
2. Iv = v for all v ∈ C
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
, · · · ,
matrix which maps each
to itself must be
(since the columns of a matrix
are the images of the basis vectors)
To show that there are 2 orbits, we know that
. Also, as
0 ⇔ v
forms a singleton orbit. Then given any
w ∈ C
, there is a matrix
A ∈ GL
) such that
(cf. Vectors and Matrices).
(R) acts on R
(C) acts on M
(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
(C): A is conjugate to a matrix of one of these forms:
, with λ 6= µ, i.e. two distinct eigenvalues
, i.e. a repeated eigenvalue with 2-dimensional eigenspace
, 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.