2
$\begingroup$

Consider any matrix $A \in \mathbb R^{n \times n}$ with the p-norm ||A||_{p} < 1.

  1. I would like to show that $\lim\limits_{k \rightarrow \infty}{A^k} = 0$.
  2. Consider the reverted scenario. Let $\lim\limits_{k \rightarrow \infty}{A^k} = 0$,

I actually have two questions regarding number 1. It's not stated which p-norm, does this mean it has to be valid for all p-norms? And then I'm a little lost on to how to prove this. Does this mean all elements of the matrix are smaller than 1?

For number 2 I have a hunch it's false. Consider a matrix with $1$ on the top right corner, $A^k$ will be $0$ but it's $p=1$ norm is 1. (and therefore also all other norms?)

Many thanks in advance!

  • 0
    A useful (standard) result is that the limit is $0$ iff all eigenvalues lie in the open unit ball of $\mathbb{C}$. Another useful result is that all eigenvalues lie in the open unit ball of $\mathbb{C}$ iff there exists an induced norm of value less than 1.2012-04-24

1 Answers 1

3

From $||A^k||_{p} \le ||A||_{p}^k=c^k$, with c<1. We know $ \lim\limits_{k \rightarrow \infty}{\|A^k\|_p} =0$, implying what you want.

  • 0
    well that was easy! I didn't really remember/think showing the norm is 0 means the matrix is also 0. thanks guys!2012-04-24