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Showing something satisfies Inner Product - Involves Orthogonal Matrices

  1. Oct 15, 2014 #1
    1. The problem statement, all variables and given/known data

    Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal.
    Show that the product <x, y> A = x · Ay is an inner product for R3.

    2. Relevant equations
    None

    3. The attempt at a solution

    I've shown that x · Dy is an inner product. I know that Z-1 is equal to ZT. I believe that will lead me somewhere. I'm just having trouble showing the property <x, x> ≥ 0. I also know that (ATx)⋅x = x ⋅ Ax.

    Just missing one step. I don't know what it is.
     
    Last edited: Oct 15, 2014
  2. jcsd
  3. Oct 15, 2014 #2

    vela

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    Hint: ## (Zx)^T = x^T Z^T ##
     
  4. Oct 15, 2014 #3
    Doing that you'll end up with (ATx)⋅x right?
     
    Last edited: Oct 15, 2014
  5. Oct 15, 2014 #4

    Dick

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    x.y=x^Ty. That turns a dot product into a matrix product. Add that to the list of clues.
     
  6. Oct 15, 2014 #5

    RUber

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    Don't neglect that D is all positive. So ##Z^T D Z## should also be non-negative, right?
     
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