Past exam of the mathematical methods I course of the University of Oxford / 2015 / 13 / a / iii / Solution

For a diagonalizable matrix, write $M=P\mathrm{diag}({\lambda }_j)P^{-1}$. Relative to an existing fixed point,Convergence to the same fixed point for every initial vector therefore holds exactly whenHere $\rho (M)$ is the spectral radius. Necessity follows by choosing an initial displacement in each eigenspace; sufficiency follows because every displayed scalar power tends to zero. The condition also makes $I-M$ an invertible matrix, so that fixed point exists uniquely for every $c$.

If instead different initial vectors may converge to different constant solutions, an eigenvalue $1$ is allowed: all other eigenvalues must have modulus less than one, and the component of $c$ in the eigenspace for $1$ must vanish. That component of the initial vector then remains unchanged. No other unit-modulus eigenvalue is compatible with convergence for arbitrary initial vectors.