Invertible linear transformation

  1. 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.
     
  2. jcsd
  3. What would happen to T-I, if 0 was an eigenvalue of T? Is it compatible with the hypothesis?
     
  4. if 0 was an eigenvalue of T then T would be singular..
     
  5. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    Ok, so if T is not invertible then Tv=0 for some v. So v corresponds to what eigenvalue of T-I?
     
  6. 0 = λ*v + I*v
    => -1 = λ

    but I know my spectral radius is <1 so contradiction...
     
  7. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    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.
     
  8. 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...
     
  9. For any norm [tex]\left\|v\right\|=\left\|-v\right\|[/tex]. Regarding the limit, remember the form of the geometric series.
     
  10. 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.
     
  11. 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...
     
    Last edited: Jan 28, 2010
  12. 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?
     
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