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Induced norms

  1. Apr 10, 2010 #1
    Let T be any square matrix and let [tex]\left\| \cdot \right\|[/tex] denote any induced norm. Prove that

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

    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

    [tex] \left\| T^{n} \right\| = sup_{x \neq 0} \frac{\left\| T^{n} x\right\|}{\left\| x \right\|} [/tex]

    But I don't know how this information helps with the solution of the problem.
  2. jcsd
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