1. The problem statement, all variables and given/known data If T is a linear transformation on R^n with || T-I || < 1, prove that T is invertible. 3. The attempt at a solution So a linear transformation T is invertible iff the matrix T is not singular. and I know for any matrix A, ||A|| > spectral radius(A). so, spectral radius(T-I) < 1.
Yes. That's it. You could also say (T+I)v=(-v) means ||T+I||>=1 and not even say anything about spectral radius. Still a contradiction with ||T+I||<1.
I need to show: sum from k=0 to infinity of (I-T)^k converges absolutely to T^(-1) so if ||T-I|| <1 then is ||I-T|| < 1? and all the properties I listed carry over? I'm still not too sure where to go with this. when the spectral radius is <1, the higher powers of the matrix tend to 0, so it clearly converges...
For any norm [tex]\left\|v\right\|=\left\|-v\right\|[/tex]. Regarding the limit, remember the form of the geometric series.
In fact, it's easier if you consider a matrix [tex]S[/tex], with [tex]\left\|S\right\|<1[/tex] and prove that: [tex]\sum_{n=0}^{\infty}S^n[/tex] Converges absolutely and compute the limit.
There is a thm that says if spectral norm <1 then A^n -> 0 as n-> infinity. and I proved above that spectral norm is <1 so I'm lost again...
What can you say about the real series: [tex] \sum_{n=0}^{\infty}\left\|S\right\|^n [/tex] When [tex]\left\|S\right\|<1[/tex]? Does it converge? if yes, what's the sum? Is it related to ypur original series if S = I-T?