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Linear Algebra -- Projection matrix question

  1. Apr 25, 2016 #1
    1. The problem statement, all variables and given/known data
    Let A be an n×n matrix which has the property that A^2 =A.
    (i) Write down the most general polynomial in A


    2. Relevant equations


    3. The attempt at a solution
    My biggest problem is that I don't even understand what the question is asking

    Is it just sum (alphaA^n)=0

    but A^n=A

    sum(alpha A)=0 ?

    I know its not the equation to find the eigenvalues as that follows, and I'm fine with that

    Av=pv where p are the eigenvalues, and v the corresponding eigenvectors
    A^2v=Apv
    Av=p(Av)
    pv=p^2v
    v(p)(p-1)=0 and hence p= 0 or 1

    But I just don't understand the first bit

    Many thanks in advance :)
     
  2. jcsd
  3. Apr 25, 2016 #2

    Simon Bridge

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    The absolute most general polynomial, degree N, in some arbitrary A would be ##P_N(A)=\sum_{n=0}^N a_nA^n## wouldn't it?
    What is special about this polynomial?
     
  4. Apr 25, 2016 #3
    Well since A is a projection matrix, surely that would imply than ##A^n=A## and hence ##P_N(A)=\sum_{n=0}^N a_nA^n## -> ##P_N(A)=\sum_{n=0}^N a_nA##
     
  5. Apr 25, 2016 #4

    Simon Bridge

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    i.e. the most general polynomial is 1st order. Well done.
     
  6. Apr 25, 2016 #5

    LCKurtz

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    Wouldn't you want to include ##A^0 = I##?
     
  7. Apr 26, 2016 #6
    Yes, yes I would ;) Thank you
     
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