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Matrix, minimal polynomial

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data

    A n × n-matrix A satisfies the equation A2 = A.

    (a) List all possible characteristic polynomials of A.

    (b) Show that A is similar to a diagonal matrix

    2. Relevant equations



    3. The attempt at a solution

    A2 = A
    so, A2 - A = 0
    A(A-I) = 0

    Our minimal polynomial is x2 - x = m(x)

    Our eigenvalues are 0 and 1, and since in our minimal polynomial each one has a multiplicity of 1, A is similar to a diagonal matrix consisting of n jordan blocks with either 1's or 0's on the diagonal and 0 everywhere else since we have an nxn matrix and n jordan blocks.

    So I've proved part (b) first.

    For part (a) we don't know how many times each eigenvalue occurs in the characteristic polynomial, so p(t) = ta(t-1)b
    where a,b=0,1,2,...,n.


    Is this correct?
     
    Last edited: Aug 25, 2011
  2. jcsd
  3. Aug 25, 2011 #2

    micromass

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    All is correct, except

    AB=0 does not imply A=0 or B=0. For example

    [tex]\left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right)[/tex]

    is a matrix A that satisfied [itex]A^2=A[/itex], but which is not 0 or I.
     
  4. Aug 25, 2011 #3
    Okay, thanks! :smile:
     
  5. Aug 25, 2011 #4

    micromass

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    :blushing: It's not my day today. Thanks ILS...
     
  6. Aug 25, 2011 #5

    I like Serena

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    No, I just deleted my post.
    They're asking for all characteristic polynomials, not for all minimal characteristic polynomials.
     
  7. Aug 25, 2011 #6

    micromass

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    No, you were correct. If a matrix satisfies [itex]A^2=A[/itex], then its minimal polynomial is not necessary [itex]x^2-x[/itex]. I should have pointed that out.

    That said, this doesn't change anything about the solutions of the OP. They remain correct.
     
  8. Aug 26, 2011 #7

    HallsofIvy

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    If A^2- A= A(A- I)= 0, then, whatever the characteristic equation of A is, it must include factors of x and x- 1.
     
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