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Homework Help: Inverse of outer-product

  1. Jul 8, 2010 #1
    Hi All,

    This is not a homework or coursework question. Yet I have a curiosity.

    If a matrix [tex]A[/tex] is an outer-product of a vector [tex]v[/tex] as : [tex]A = v v^{\top}[/tex]

    Then can [tex]A^{-1}[/tex], inverse of [tex]A[/tex], be also expressed as an outer-product of some other vector?

    Please point me how to approach the question, how to find the answer or how to say it is possible or not.

    Thanks.
     
  2. jcsd
  3. Jul 8, 2010 #2

    Hurkyl

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    Staff Emeritus
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    Gold Member

    If A is 0x0 or 1x1, inverting it is easy. Otherwise...

    Exercise: If v is nonzero, prove that A has rank 1.


    You can do better:

    Exercise: A matrix has rank 1 if and only if it is the outer product of two nonzero vectors.
     
  4. Jul 8, 2010 #3
    oops!! Thanks.
     
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