Invariant Subspaces - Eigenvectors

Question

I have the following function: $\rho : C_4 \rightarrow GL_2(\Bbb C)$

$\rho(e) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} $

$\rho(x) = \begin{pmatrix} 0 & i \\ i & 0 \\ \end{pmatrix} $

$\rho(x) = \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix} $

$\rho(x) = \begin{pmatrix} 0 & -i \\ -i & 0 \\ \end{pmatrix} $

I have shown that $(\Bbb C^2,\rho)$ is a representation, and I have found eigenvalues and eigenvectors for all four matrices.

Each matrix has the same two eigenvectors:

$\ \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} $ or $\ \begin{pmatrix} 1 \\ -2 \\ \end{pmatrix} $

Now, I have to find a $1$-dimensional subspace of $(\Bbb C^2,\rho)$, but my two eigenvectors are linearly independent, so I'm a bit stuck, because this would give me a $2$-dimensional subspace.

Answer 1

Well, the matrices $\rho(e)$ and $\rho(x^2)$ are just $I$ and $-I$ where $I$ is the identity matrix, so these have all vectors as eigenvectors.

For $\rho(x)$ and $\rho(x^3)$, you probably wanted to mean the vectors $v_1=\pmatrix{1\\ 1}$ and $v_2=\pmatrix{1\\-1}$.

A one dimensional invariant subspace must be generated by an eigenvector.

So, now we have two pieces of one dimensional $\rho$-invariant subspaces:
$\quad{\rm span}(v_1)\ $ and $\ {\rm span}(v_2)$.