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Idempotent matrix problem - HELP

  1. Oct 15, 2009 #1
    idempotent matrix problem - HELP :(

    Hi, I have the following problem with an idempotent matrix and I am stuck...

    If A is an idempotent (A^2 = A), is E - nA invertible /E is the identity matrix and n is a real number/ and why?

    I've tried with setting (E - nA)(E - nA) but it doesn't get me anywhere..
    My instinct is telling me there it should be invertible in a lot of the cases, but... :/
     
  2. jcsd
  3. Oct 15, 2009 #2

    Dick

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    Re: idempotent matrix problem - HELP :(

    If (E-nA) were NOT invertible, then there would be a nonzero vector v such that (E-nA)v=0. Where does that lead you?
     
  4. Oct 15, 2009 #3
    Re: idempotent matrix problem - HELP :(

    I am sorry, I am quite new to Linear Algebra I am not really sure what the answer to your question should be... Can you explain a little bit more why there should be a non-zero vector such that (E-nA)v=0?

    Thank you for helping me out! You're a life saver!
     
  5. Oct 15, 2009 #4

    Dick

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    Re: idempotent matrix problem - HELP :(

    If M is a square matrix, then M being invertible is equivalent to the kernel or null space of M being {0}. If M is NOT invertible, then the kernel of M is not {0}. Meaning there is nonzero vector v in the kernel, hence Mv=0. You must have covered this. Can you look back in your book or notes for that kind of thing? It's pretty important.
     
  6. Oct 15, 2009 #5
    Re: idempotent matrix problem - HELP :(

    I checked my notes and as far as I understand you're talking about the invertible matrix theoremy...
    Can we then say that:
    Assuming that E-nA is not invertible means that there will be a non-zero vector x, such that (E-nA)x=0. However, this suggests that E-nA is not row equivalent to E which then implies that E-nA is not a product of elementary matrices. This implies that E-nA is not invertible???

    I'm sorry I'm such a pain but I am really lost...

    Thanks again..
     
  7. Oct 15, 2009 #6

    Dick

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    Re: idempotent matrix problem - HELP :(

    You are making it much too hard. If (E-nA)v=0 then Ev-nAv=0. What is Ev?
     
  8. Oct 15, 2009 #7
    Re: idempotent matrix problem - HELP :(

    Hm..

    Well, Ev = nAv => v = nAv .. Doesn't this mean that nA=1?
     
  9. Oct 15, 2009 #8

    Dick

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    Re: idempotent matrix problem - HELP :(

    NO! You can't divide both sides by a vector!!! But it does mean Av=v/n. Now operate on both sides with A again remembering A is idempotent.
     
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