quasi
Posts:
12,052
Registered:
7/15/05


Re: Question about linear algebra matrix pnorm
Posted:
Jan 8, 2013 1:36 AM


rxjwg98@gmail.com wrote:
>Hi, >I am reading a book on matrix characters. It has a lemma on >matrix pnorm. I do not understand a short explaination in >its proof part. > >The Lemma is: If F is Rnxn and Fp<1 (pnorm of F), then >IF is nonsingular.... > >In its proof part, it says: Suppose IF is singular. It >follows that (IF)x=0 for some nonzero x. But then >xp=Fxp implies Fp>=1, a contradiction. Thus, IF >is nonsingular. > >My question is about how it gets: >But then xp=Fxp implies Fp>=1 > >Could you tell me that? Thanks a lot
It's an immediate consequence of the definition of the matrix pnorm. By definition,
<http://en.wikipedia.org/wiki/Matrix_norm>
Fp = max (Fxp)/(xp)
where the maximum is taken over all nonzero vectors x.
Thus, Fp < 1 implies
(Fxp)/(xp) < 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 => Fxp = xp => (Fxp)/(xp) = 1,
contradiction.
quasi

