We have the following stationary matrix iteration

$$x_{k+1} = Mx_k + c$$

where $M$ is nxn matrix and $c$ is a vector.

Let $r(M)$ denote the spectral radius of $M$.

Show that spectral radius $r(M)$ is infimum over all compatible matrix norms induced by vector norms

Then using this show that

IF $x^\ast = \lim x_k$ for any $x_0$

then $r(M) < 1$

where $x^\ast$ is the fixed point of the iteration.

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This kind of stuffs are pretty standard. Have you ever looked up any textbook or reference book for a proof? –  user1551 Jan 1 '13 at 21:48