We are doing norms in my numerical analysis class, and I'm having a lot of trouble. We have to prove
||x^(k) - x|| <= ||T||^(k)*||x^(1) - x^(0)||/(1 - ||T||), where T is a nxn matrix and ||T||<1.
He gave us a hint, and so so far I have it down to
<= ||T||^(k)(||x^(1) - x^(0)|| + ||x^(1) - x||)
but I'm really not sure what to do next. Can anyone help? And maybe explain as you go, because I'm very new at this stuff, and my book doesn't have much on this topic. Also, do you know any sites that talk about it?
I know x = Tx + c and x^(k) = Tx^(k-1) + c but that's about it.