 Solution

A fixed point obeys $(I-M)X=c$. Thus the necessary and sufficient existence condition isIf $1$ is not an eigenvalue of $M$, then $I-M$ is an invertible matrix and the unique constant solution is $\boxed{X=(I-M{)}^{-1}c}$. If $I-M$ is singular but the equation is consistent, all constant solutions are $X_0+\ker (I-M)$ for any particular solution $X_0$. For an existing fixed point, $x_{k+1}-X=M(x_k-X)$; the successive difference in the question likewise obeys $\delta x_{k+1}=M\delta x_k$.