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Topic: Question about linear algebra matrix p-norm
Replies: 6   Last Post: Jan 9, 2013 2:42 AM

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quasi

Posts: 10,232
Registered: 7/15/05
Re: Question about linear algebra matrix p-norm
Posted: Jan 8, 2013 1:36 AM
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rxjwg98@gmail.com wrote:

>Hi,
>I am reading a book on matrix characters. It has a lemma on
>matrix p-norm. I do not understand a short explaination in
>its proof part.
>
>The Lemma is: If F is Rnxn and |F|p<1 (p-norm of F), then
>I-F is non-singular....
>
>In its proof part, it says: Suppose I-F is singular. It
>follows that (I-F)x=0 for some nonzero x. But then
>|x|p=|Fx|p implies |F|p>=1, a contradiction. Thus, I-F
>is nonsingular.
>
>My question is about how it gets:
>But then |x|p=|Fx|p implies |F|p>=1
>
>Could you tell me that? Thanks a lot


It's an immediate consequence of the definition of the matrix
p-norm. By definition,

<http://en.wikipedia.org/wiki/Matrix_norm>

|F|p = max (|Fx|p)/(|x|p)

where the maximum is taken over all nonzero vectors x.

Thus, |F|p < 1 implies

(|Fx|p)/(|x|p) < 1 for all nonzero vectors x,

But if I - F was singular, then, as you indicate, F would have
a nonzero fixed point x, say.

Then

Fx = x
=> |Fx|p = |x|p
=> (|Fx|p)/(|x|p) = 1,

contradiction.

quasi



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