Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix? If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)?
I always thought that there was an invariant plane if all 3 equations were the same when trying to find the eigenvectors, but does this only happen when there is a repeated eigenvalue, or does it happen also when there are 3 distinct eigenvalues? 

For any $n \times n$ (complex) matrix $A$, there are always invariant subspaces of $\mathbb C^n$  of all dimensions $\le n$.  If $A$ is upper triangular, the $k$-dimensional subspace spanned by the first $k$ standard unit vectors is invariant.  And every square matrix is similar to an upper triangular matrix.
