Can anyone help me with with this proof???
Show that a sufficient condition for...
Lim as k-> infinity, A^K = 0
is
||A|| < 1 (NOTE: this is the 2-norm)
any help would be greatly appreciated...
|| A^K ||_2 = max_{x !=0} [||A^K x||_2 / ||x||_2]
................ = max_{x!=0}|| [A ( A^{K-1} x||_2 / ||X||_2]
................ < k max_{x!=0}|| [ A^{K-1} x||_2 / ||X||_2]
for some k such that: ||A||_2 < k < 1, hence:
|| A^K ||_2 < k^K
But as K -> infty, k^K ->0, so
Lim as k-> infinity, ||A^K||_2 = 0
hence A^K -> 0 (as in this case convergence in norm implies convergence elementwise
though I'm not sure if you will have to prove this).
RonL