Invertible linear transformation

[CarmineCortez]

Homework Statement

If T is a linear transformation on R^n with || T-I || < 1, prove that T is invertible.

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.

 

[JSuarez]

What would happen to T-I, if 0 was an eigenvalue of T? Is it compatible with the hypothesis?

 

[CarmineCortez]

if 0 was an eigenvalue of T then T would be singular..

 

[Dick]

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?

 

[CarmineCortez]

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

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

 

[Dick]

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.

 

[CarmineCortez]

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...

 

[JSuarez]

For any norm $$\left\|v\right\|=\left\|-v\right\|$$. Regarding the limit, remember the form of the geometric series.

 

[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.

 

[CarmineCortez]

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...

 

[JSuarez]

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?