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Dimension of space composed of powers of a matrix

  1. Nov 29, 2011 #1
    Let A be an n*n matrix.
    Consider the space [itex]span \{ I, A, A^2, A^3, ... \} .[/itex]
    How would one show that the dimension of the space never exceeds n?
    I feel like the answer lies somewhere near the Cayley-Hamilton theorem, but I can't quite grasp it.
     
  2. jcsd
  3. Nov 30, 2011 #2

    micromass

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    The answer is indeed given by Cayley-Hamilton.

    Basically, what you must prove is that every matrix like An, An+1, etc. can be written as a linear combination of the matrices I,A,...,An-1.

    Now, what does Cayley-Hamilton say?? Doesn't this theorem answer our question?

    How would you use Cayley-Hamilton to write An as a linear combination of I,A,...,An-1?
     
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