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Homework Help: Show companion matrix is similar to the following matrix

  1. Feb 20, 2014 #1
    1. The problem statement, all variables and given/known data
    need to show companion matrix is similar to the following matrix
    F552C6O.jpg (here is the picture of the matrix)

    2. Relevant equations

    here is the companion matrix

    information on matrix similarity

    3. The attempt at a solution

    say given matrix is A and companion matrix is C then need to show

    A = P^-1 * C * P for some invertible matrix P

    i guess i could reduce guesswork by rewriting as

    P*A = C*P

    but even then it does not seem to be the ideal way to go about things.
    Last edited: Feb 20, 2014
  2. jcsd
  3. Feb 20, 2014 #2
    I don't think you put in the right link to the matrix...
  4. Feb 20, 2014 #3
    oh lol fixed
  5. Feb 20, 2014 #4
    What is the characteristic and minimal polynomial of that matrix? Can you construct the rational canonical form based off of elementary divisors?
  6. Feb 20, 2014 #5
    i know char for companion but that is all.

    i also know:

    if same minimal poly then similar

    if same frobenius canonical form then similar

    but no clue how to go about finding them
  7. Feb 20, 2014 #6
    Since your matrix is upper triangular, finding the characteristic polynomial is easy: It is just ##∏ (x-\lambda_i)##. If you can show that this is the minimal polynomial as well, then you are done.
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