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Relationship between two matrices

  1. Apr 26, 2014 #1
    Hello. I need some help with one question about relationship of two matrices.
    The task:
    Suppose that I is identity matrix, u - is vector, u' is transposed vector, α - real number. It can be prove that inverse matrix of I+α*u*u' has similar form I+x*u*u'. The task is to find x.

    I tried to calculate inverse of this matrix I+α*u*u', but first of all when i dont know dimension it is very difficult to calculate inverse matrix. I tried to take example when u is from R^2, but even then i cant calculate x.

    Any help would be appreciate. Any hints or something else.
     
  2. jcsd
  3. Apr 26, 2014 #2
  4. Apr 26, 2014 #3

    Matterwave

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    Gold Member

    Instead of trying to calculate the inverse, why not assume the inverse is like I+x*u*u' and see what happens when you take the product?

    $$(I+\alpha uu^T)(I+xuu^T)=?$$
     
  5. Apr 27, 2014 #4
    ok i will try. thanks for help, Matterwawe!
     
  6. Apr 27, 2014 #5
    $$(I+\alpha uu^T)(I+xuu^T)=I$$
    $$(I+I*xuu^T+I*\alpha uu^T+\alpha uu^T*xuu^T=I$$
    So after that
    $$I*xuu^T+I*\alpha uu^T+\alpha uu^T*xuu^T=0$$
    $$xuu^T+\alpha uu^T+\alpha uu^T*xuu^T=0$$
    $$(x+\alpha) uu^T+\alpha*x (uu^T*uu^T)=0$$
    $$(x+\alpha) uu^T=-\alpha*x (uu^T*uu^T)$$

    And whats next? How can i express x?

    $$(x+\alpha)/-\alpha*x =(uu^T)^{-1}(uu^T*uu^T)$$
    $$-1/\alpha-1/x=(uu^T)^{-1}(uu^T*uu^T)$$
    $$-1/x=(uu^T)^{-1}(uu^T*uu^T)+1/\alpha$$
     
    Last edited: Apr 27, 2014
  7. May 2, 2014 #6
    From equation
    $$xuu^T+\alpha uu^T+\alpha uu^T*xuu^T=0$$
    follows with
    $$uu^T*xuu^T = xu(u^Tu)u^T = x (u^Tu) uu^T$$ (u^Tu is a scalar)
    the solution of your question:
    $$x = -\alpha / (1+\alpha u^Tu)$$
     
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