Ah… that "diagonalizable" is doing some heavy lifting there! I was wondering how exactly you’re going to make, say, a rotation matrix to converge anything to anything that’s not already an eigenvector. And rotation matrices certainly aren’t degenerate! Though apparently non-diagonalizable matrices can be called defective which is such a dismissive term :( Poor rotation matrices, why are they dissed so?!
Love them, those rotation matrices.
Take logarithm of the eigenvalues and you get back the angle. This to me had solidified the notion that angles are essentially a logarithmic notion ... Made more rigorous by the notion of exponential maps