I have a matrix which I have found the eigenvalues for.
$$\begin{bmatrix}1/4 & 1/2 & 1/2 & 1/2\\1/4 & 1/2 & 0 & 0\\1/4 & 0 & 1/2 & 0\\1/4 & 0 & 0 & 1/2\end{bmatrix}$$
Two of the eigenvalues are $1/2$. But when I try to find the eigenvectors associated with that eigenvalue I get the following system of equations.
$$x_2 + x_3 + x_4 = \frac 12 x_1$$ $$x_1 = 0$$ $$=> x_2 + x_3 + x_4 = 0$$
So it looks as though I have an eigen-plane instead of an eigenvector. Is that possible, or am I misinterpreting the result?
Yes, that is perfectly possible. To each eigenvalue there corresponds an eigenspace that can have any dimension. You have an eigenvalue that has some algebraic multiplicity and some geometric multiplicity as well. The geometric multiplicity (the dimension of the eigenspace) is always less or equal to the algebraic multiplicity.