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Homework Help: Complex matrix space:

  1. Oct 3, 2007 #1
    1. The problem statement, all variables and given/known data
    If C is a n by n matrix with complex coefficient show that there is an integer k >0 that depends only on n such that C, C^2,C^3,...C^k are lin dependant.

    3. The attempt at a solution

    Its meant to be a full proof but the only idea I vaguely have is that i^4=i. I'm sure this is something standard to prove but I couldn't find it on google, so a link with an explanation would be appreciated.
  2. jcsd
  3. Oct 3, 2007 #2
    I can't remember much linear algebra but I would try considering the following. Suppose the dimension of a vector space is 6. Is it possible to extract 7 linearly independent vectors from that vector space?
  4. Oct 3, 2007 #3


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    Science Advisor
    Homework Helper

    Cayley-Hamilton tells us that C satisfies its own characteristic polynomial.
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