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Jordan form of f^​2 and f^​3 knowing that m. polynomial of f is x^​7

  1. Jul 16, 2013 #1
    The problem statement
    Let V be a vector space of dimension 8 and f (endomorphism) such that the minimal polynomial of f is x^​7. If B={v1,...,v8} is the Jordan basis of f, find the Jordan form and a Jordan basis for f^​2 and f^​3.

    The attempt at a solution

    Ok, I am having some trouble to solve this. My ideas are:

    The minimal of f is x^​7, then f is nilpotent of index 7. The only eigenvalue of the Jordan form is 0 and the multiplicity of 0 as a root of the minimal tells me the biggest size of the jordan blocks, in this case is a 7x7 block. As the Jordan matrix is an 8x8 matrix, then, it can only have a 7x7 block and a 1x1 block. Now, if the minimal of f is x^​7, then the minimal of f^​2 and of f^​3 have to be x^​4 and x^​3 respectively. So, the biggest blocks in f^​2 and f^​3 are of size 4x4 and 3x3 respectively.

    Now, I don't know how to get more information about the blocks of each Jordan form and I don't have a clue about the Jordan basis. I would appreciate any help. And sorry if I made any grammar or expression mistakes, english is not my native language. Thanks!
     
  2. jcsd
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