= Solution
For a <diagonalizable matrix>, write $M=P\operatorname{diag}(\lambda_j)P^{-1}$. Relative to an existing <fixed point>,
$$
x_k-X=P\operatorname{diag}(\lambda_j^{k-1})P^{-1}(x_1-X).
$$
Convergence to the \b[same fixed point for every initial vector] therefore holds exactly when
$$
\boxed{|\lambda_j|<1\text{ for all }j,\quad\text{equivalently }\rho(M)<1.}
$$
Here $\rho(M)$ 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.
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