Just a quick Q -- I know how to find the characteristic polynomial, solve for the eigenvalues, and find the eigenvectors from there, but I was wondering, say you have a double eigenvalue (like your polynomial reduces to (x-5)(x-5) -- then you would have a double eigenvalue of 5)...is it then possible to determine a basis of eigenvalues for a 2x2 matrix? Or since you only have one eigenvector, can you not determine a basis?