Show that the matrix representation of the dihedral group D4 by M is irreducible.

[blueyellow]

1. Homework Statement [/b]

Show that the matrix representation of the dihedral group D4 by M is irreducible.

You are given that all of the elements of a matrix group M can be generated
from the following two elements,

A=
|0 -1|
|1 0|

B=
|1 0|
|0 -1|

in the sense that all other elements can be written A^n B^m for integer m, n >or= 0.
Find the remaining elements in M.

The Attempt at a Solution

I tried reading through the notes and they say:
An n-dimensional matrix REP M(G) of a finite group G is reducible if there exists a similarity transformation S such that

S^(-1) M (g) S=
|M(subscript 1) (g) 0 |
|0 M(subscript 2) (g)|

for each g (is an element of G)

but I do not know how I would go about starting with trying to find a similarity transformation

 

[micromass]

Hi blueyellow!

Maybe it's best to start with the second part and find all the elements of M.

Anyway, you have to show that the representation is not reducible. That is, you need to show that there is no basis such that A and B can simultaniously be diagonalized. Do this by showing that the eigenspaces of the matrices do not coincide.