Results 1 to 8 of 8

Thread: Square matrix and induced norm

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    98

    Square matrix and induced norm

    Let T be any square matrix and let $\displaystyle \left\| \cdot \right\|$ denote any induced norm. Prove that


    $\displaystyle lim_{n \rightarrow _{\infty}} \left\| T^{n} \right\| ^{1/n} $ exists and equals $\displaystyle inf _{n=1,2,\cdots } \left\| T^{n} \right\| ^{1/n} $


    I am not sure how I go about proving that the limit exists.
    For the infimum, I think it has something to do with the fact that


    $\displaystyle \left\| T^{n} \right\| = sup_{x \neq 0} \frac{\left\| T^{n} x\right\|}{\left\| x \right\|} $
    But I don't know how this information helps with the solution of the problem.
    Last edited by math8; Apr 11th 2010 at 09:52 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    This isn't an answer; I'm just retyping your equations which are broken for me (hotlinking I think)...

    Quote Originally Posted by math8 View Post
    Let T be any square matrix and let $\displaystyle ||\cdot||$ denote any induced norm. Prove that $\displaystyle \lim_{n\to\infty}||T^n||^{1/n}$ exists and equals $\displaystyle \inf_{n=1,2,\cdots} ||T^n||^{1/n}$.

    I am not sure how I go about proving that the limit exists.
    For the infimum, I think it has something to do with the fact that

    $\displaystyle ||T^n|| = \sup_{x\neq0} {||T^nx||\over||x||}$

    But I don't know how this information helps with the solution of the problem.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by maddas View Post
    This isn't an answer; I'm just retyping your equations which are broken for me (hotlinking I think)...
    Originally Posted by math8
    Let T be any square matrix and let $\displaystyle \|\cdot\|$ denote any induced norm. Prove that $\displaystyle \lim_{n\to\infty}\|T^n\|^{1/n}$ exists and equals $\displaystyle \inf_{n=1,2,\ldots} \|T^n\|^{1/n}.$
    This is Gelfand's spectral radius formula (proof in the link).

    [Thanks to Maddas for making the original post readable.]
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Feb 2009
    Posts
    98
    Thanks to Maddas and to Opalg.

    I can understand the proof of the existence of $\displaystyle lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$.

    $\displaystyle lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$ exists and is equal to $\displaystyle \rho (T) $, the spectral radius of T.

    But how do we show that $\displaystyle \rho (T) = inf_{n=1,2, \cdots } \left\| T^{n} \right\| ^{1/n}$ ?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by math8 View Post
    Thanks to Maddas and to Opalg.

    I can understand the proof of the existence of $\displaystyle lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$.

    $\displaystyle lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$ exists and is equal to $\displaystyle \rho (T) $, the spectral radius of T.

    But how do we show that $\displaystyle \rho (T) = inf_{n=1,2, \cdots } \left\| T^{n} \right\| ^{1/n}$ ?
    This follows from the fact the norm has to be submultiplicative, so that $\displaystyle \|S^2\|\leqslant\|S\|^2$ for any matrix S.

    Suppose that, for some n, $\displaystyle \|T^n\|^{1/n}=\lambda<\rho(T)$. Then $\displaystyle \|T^n\|=\lambda^n$, $\displaystyle \|T^{2n}\| = \|(T^n)^2\| \leqslant \|T^n\|^2= \lambda^{2n}$, and by induction $\displaystyle \|T^{2^kn}\|\leqslant\lambda^{2^kn}$ for all k.

    Therefore $\displaystyle \|T^{2^kn}\|^{1/(2^kn)}\leqslant\lambda$. let $\displaystyle k\to\infty$ to see that $\displaystyle \rho(t)\leqslant\lambda$ — contradiction.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Feb 2009
    Posts
    98

    Square matrix and induced norm

    I might be wrong but according to that proof, does this just mean that $\displaystyle \rho (T)$ must be a lower bound for $\displaystyle \{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$ ?

    How do we show that $\displaystyle \rho (T)$ is actually the greatest lower bound? I mean is there an n such that $\displaystyle \rho (T) = \left\| T^{n} \right\| ^{1/n} $ ?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by math8 View Post
    I might be wrong but according to that proof, does this just mean that $\displaystyle \rho (T)$ must be a lower bound for $\displaystyle \{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$ ?

    How do we show that $\displaystyle \rho (T)$ is actually the greatest lower bound? I mean is there an n such that $\displaystyle \rho (T) = \left\| T^{n} \right\| ^{1/n} $ ?
    Yes, it shows that $\displaystyle \rho (T)$ must be a lower bound for $\displaystyle \{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$. But I thought you had already shown that $\displaystyle \rho (T) = \lim_{n\to\infty} \left\| T^{n} \right\| ^{1/n} $. Taken together, those results show that $\displaystyle \rho (T)$ is the greatest lower bound.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Feb 2009
    Posts
    98
    Oh, I didn't realize that. I actually thought that for that result, one actually has to show that there is an n such that $\displaystyle \rho (T) = \left\| T^{n} \right\| ^{1/n} $. (without the limit)

    But I guess you're right.

    Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inner Product space not induced by L1 norm
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 23rd 2010, 12:28 AM
  2. How to compute Induced Matrix Norm?
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Mar 8th 2010, 10:44 AM
  3. induced norm
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Nov 23rd 2009, 05:58 PM
  4. [SOLVED] Induced matrix norms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 22nd 2009, 10:06 PM
  5. Vector Norm and Matrix Norm
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 18th 2008, 10:49 AM

Search Tags


/mathhelpforum @mathhelpforum