I already know the long-winded way to find the span of an eigenspace. Its to find how many invariant lines (from the matrix that go through the origin) face different directions, and the number of different lines facing different directions is the dimension of the eigenspace.
I know that if there are no repeated eigenvalues, then the eigenspace spans the same dimention, but if there are repeated eigenvalues, the eigenspace can differ (not always since having repeated eigenvalues doesnt always imply a smaller dimention for the eigenspace).
What is a quick method to find the eigenspace of a matrix because I want to be able to tell if a matrix is diagonalizable or not?