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Homework Help: Proving transpose of orthogonal matrix orthogonal

  1. Aug 9, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that if A is orthogonal, then AT is orthogonal.

    2. Relevant equations

    AAT = I

    3. The attempt at a solution

    I would go about this by letting A be an orthogonal matrix with a, b, c, d, e, f, g, h, i , j as its entries (I don't know how to draw that here)...but this would be a 3x3 matrix with entries a, b, c, d, e, f, g,h, i, j. I would then construct AT, and then multiply the two matrices. I should find that the non-diagonal entries are zero, but how would I show that the diagonal entries are 1?
  2. jcsd
  3. Aug 9, 2011 #2
    Use the fact that the columns are all orthogonal unit vectors.
  4. Aug 9, 2011 #3
    Ok. I understand how I would prove that A times the transpose of A has orthogonal columns...but how would I prove they are unit vectors?
  5. Aug 9, 2011 #4
    What definition are you using for an orthogonal matrix? Note that orthogonal unit vectors for rows and columns is equivalent to [itex] AA^T = A^TA=I[/itex]. The equivalence of these definitions is perhaps in your book or can certainly be found online.

    I would use the second definition. A is orthogonal if and only if [itex] AA^T=A^TA = I[/itex]. To show [itex]A^T[/itex] is orthogonal, make use of the fact that [itex](A^T)^T=A[/itex].
  6. Aug 9, 2011 #5
    If you're assumed that A is already orthogonal then you don't need to prove that the columns are orthogonal unit vectors. That's the definition of an orthogonal matrix, thus already being in your assumption that A is orthogonal.
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