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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
    http://en.wikipedia.org/wiki/Companion_matrix

    information on matrix similarity
    http://en.wikipedia.org/wiki/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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