Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.