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Finding the minimal polynomial of a matrix?

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Let f(x) be an irreducible polynomial cubic in Q. For example

    f(x) = ax^3 + bx^2 + cx + d

    Let A be a 3 x 3 matrix with entries in Q such that char(A,x) = f(x). Find the minimal polynomial m(x) of A. Can you generalize to a degree n polynomial?

    2. Relevant equations

    3. The attempt at a solution

    If the char(A,x) = f(x) then the companion matrix is...

    [0, 0, -d]
    [1, 0, -c]
    [0, 1, -b]

    Since the companion matrix's characteristic polynomial = its minimal polynomial, does this mean the minimal polynomial is just f(x). I'm missing something, aren't I....
  2. jcsd
  3. Mar 22, 2009 #2


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    Staff Emeritus
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    What about the a???
  4. Mar 22, 2009 #3
    Oh. The companion matrix would be...

    [0 0 -d/a]
    [1 0 -c/a]
    [0 1 -b/a]

    so m(x) = x^3 + (b/a)x^2 + (c/a)x + (d/a)?
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