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Math Help - Orthogonal Matrix Preserve Length

  1. #1
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    Orthogonal Matrix Preserve Length

    Ther question is:

    Show that if Q is orthogonal, then Q preserves length. In other words, for all vectors x, the length of x equals the length of Qx.
    --------------------------------------------

    Ok, my understanding is that an orthogonal matrix is a square matrix where P^T = P^{-1}.

    Also, the length of the column vectors or row vectors is equal to 1. At least, a theorem describes an n X n matrix P as being orthogonal if and only if the column vectors form an orthonormal set. Also, orthonormal seems to mean a set where the length equals 1 and the sets if dotted with each other will equal the zero vector.

    I guess I am wanting confirmation. Because the columns and rows have length one, if we perform matrix multiplication with a vector x as described above, we effectively multiply that vector by one, thus preserving length. I am not quite sure how to answer the question other than saying that the length of the columns and rows are 1...and going from there. THANKS!
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  2. #2
    Senior Member Shanks's Avatar
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    Notice the fact:
    the square of the norm of x is equal to x*x,( * means transposition)
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