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Math Help - Square matrix and induced norm

  1. #1
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    Square matrix and induced norm

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


    lim_{n \rightarrow _{\infty}} \left\| T^{n} \right\| ^{1/n} exists and equals  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


     \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; April 11th 2010 at 10:52 AM.
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  2. #2
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    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 ||\cdot|| denote any induced norm. Prove that \lim_{n\to\infty}||T^n||^{1/n} exists and equals \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

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

    But I don't know how this information helps with the solution of the problem.
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  3. #3
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    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 \|\cdot\| denote any induced norm. Prove that \lim_{n\to\infty}\|T^n\|^{1/n} exists and equals \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.]
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  4. #4
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    Thanks to Maddas and to Opalg.

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

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

    But how do we show that  \rho (T) = inf_{n=1,2, \cdots } \left\| T^{n} \right\| ^{1/n} ?
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  5. #5
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    Quote Originally Posted by math8 View Post
    Thanks to Maddas and to Opalg.

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

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

    But how do we show that  \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 \|S^2\|\leqslant\|S\|^2 for any matrix S.

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

    Therefore \|T^{2^kn}\|^{1/(2^kn)}\leqslant\lambda. let k\to\infty to see that \rho(t)\leqslant\lambda — contradiction.
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    Square matrix and induced norm

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

    How do we show that \rho (T) is actually the greatest lower bound? I mean is there an n such that  \rho (T) =  \left\| T^{n} \right\| ^{1/n} ?
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  7. #7
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    Quote Originally Posted by math8 View Post
    I might be wrong but according to that proof, does this just mean that \rho (T) must be a lower bound for  \{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \} ?

    How do we show that \rho (T) is actually the greatest lower bound? I mean is there an n such that  \rho (T) =  \left\| T^{n} \right\| ^{1/n} ?
    Yes, it shows that \rho (T) must be a lower bound for  \{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}. But I thought you had already shown that  \rho (T) = \lim_{n\to\infty} \left\| T^{n} \right\| ^{1/n} . Taken together, those results show that \rho (T) is the greatest lower bound.
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  8. #8
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    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  \rho (T) =  \left\| T^{n} \right\| ^{1/n} . (without the limit)

    But I guess you're right.

    Thanks a lot!
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