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Homework Help: Orthogonal Matrix

  1. Nov 28, 2006 #1
    Assume that [tex] I [/tex] is the [tex] 3\times 3 [/tex] identity matrix and [tex] a [/tex] is a non-zero column vector with 3 components. Show that:



    [tex] I - \frac{2}{| a |^{2}}aa^{T} [/tex] is an orthogonal matrix?


    My question is how can one take the determinant of [tex] a [/tex] if it is not a square matrix? Is there a flaw in this problem?

    Thanks
     
    Last edited: Nov 28, 2006
  2. jcsd
  3. Nov 28, 2006 #2

    OlderDan

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    I assume you are referring to the [tex] | a |^{2} [/tex] and I also assume that is the inner product (dot product) for the vector. It's just a normalization factor
     
  4. Nov 28, 2006 #3

    HallsofIvy

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    Yes. |a| is not a "determinant", it is the length of the vector a.
     
  5. Nov 28, 2006 #4
    Remember that [itex]aa^{T} [/itex] does NOT equal [itex]a^{T}.a[/itex], the scalar product. Use matrix multiplication. You don't need to find the determinant of anything either.
     
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