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Homework Help: A question about matrix

  1. Jul 19, 2010 #1
    1. The problem statement, all variables and given/known data

    v is a vector with norm(v)<1
    what is the inverse of (I+vv') where I is a identity matrix

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 20, 2010 #2
    I don't see how this simplifies at all. If

    [tex] A \ = \ I \ + \ vv' [/tex] then

    [tex] A_{ij} \ = \ v_iv_j [/tex] when [tex] i \neq j [/tex] and

    [tex] A_{ij} \ = \ v_iv_j + 1 [/tex] when [tex] i = j [/tex].

    I can't see how [tex] A^{-1} [/tex] can turn out pretty. But maybe I'm just missing something...it is getting pretty late.
     
  4. Jul 20, 2010 #3
    UPDATE:

    All I can seem to find is

    [tex] det(A) \ = \ ||\textbf{v}||^2 \ + \ 1. [/tex]

    I'm not 100% certain on this since I didn't really construct a fool proof but instead made a few "gut" leaps, but I think (and hope) it holds. This result is sort of pretty; however, I still stand by what I said earlier: I highly doubt [tex] A^{-1} [/tex] is pretty.
     
  5. Jul 20, 2010 #4

    hunt_mat

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    Homework Helper

    I think I saw someone say that the matrix defined by an out product has rank 1 and therefore not invertible.

    Mat
     
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