# Invertible linear transformation

by CarmineCortez
Tags: invertible, linear, transformation
 P: 33 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.
 P: 403 What would happen to T-I, if 0 was an eigenvalue of T? Is it compatible with the hypothesis?
 P: 33 if 0 was an eigenvalue of T then T would be singular..
HW Helper
Thanks
P: 25,008

## Invertible linear transformation

 Quote by CarmineCortez if 0 was an eigenvalue of T then T would be singular..
Ok, so if T is not invertible then Tv=0 for some v. So v corresponds to what eigenvalue of T-I?
P: 33
 Quote by Dick Ok, so if T is not invertible then Tv=0 for some v. So v corresponds to what eigenvalue of T-I?
0 = λ*v + I*v
=> -1 = λ

 Sci Advisor HW Helper Thanks P: 25,008 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.
 P: 33 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...
 P: 403 For any norm $$\left\|v\right\|=\left\|-v\right\|$$. Regarding the limit, remember the form of the geometric series.
 P: 403 In fact, it's easier if you consider a matrix $$S$$, with $$\left\|S\right\|<1$$ and prove that: $$\sum_{n=0}^{\infty}S^n$$ Converges absolutely and compute the limit.
P: 33
 Quote by JSuarez In fact, it's easier if you consider a matrix $$S$$, with $$\left\|S\right\|<1$$ and prove that: $$\sum_{n=0}^{\infty}S^n$$ 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...
 P: 403 What can you say about the real series: $$\sum_{n=0}^{\infty}\left\|S\right\|^n$$ When $$\left\|S\right\|<1$$? Does it converge? if yes, what's the sum? Is it related to ypur original series if S = I-T?

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