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Show that A is an orthogonal matrix

  1. Jul 22, 2010 #1
    1. The problem statement, all variables and given/known data



    If {aj} and {bj} are two separate sets of orthonormal basis sets, and are related by

    ai = [tex]\sum[/tex]jnAijbj

    Show that A is an orthogonal matrix

    2. Relevant equations

    Provided above.





    3. The attempt at a solution

    Too much latex needed to show what I tried, but basically I considered the properties of an orthogonal matrix: AAT = I and considered which elements would multiply and sum to give the 1's and 0's of the identity matrix.

    I then considered the fact that aj.ak = o if j and k are different and 1 if they are the same, and the same for the vectors bj.

    Then I thought of the property of preserving norms, but can't see how to connect it to this problem.
     
  2. jcsd
  3. Jul 22, 2010 #2

    HallsofIvy

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    Staff Emeritus
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    More to the point is that an orthonormal matrix is a matrix whose columns (or rows), thought of as vectors, are orthonormal- that is, each has length 1 and any two are orthogonal.
     
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