1. ## Optimization proof...

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...

2. Originally Posted by ginafara
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

3. beautiful... thanks for the help... I was on the right track, but wasn't confident in my work.

beautiful proof... thanks again