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Math Help - Orthogonal Matrix proof

  1. #1
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    Orthogonal Matrix proof

    An n x n matrix Q is said to be an orthogonal matrix if the column vectors of Q form an orthornormal minimal spanning set of R^n. Prove the following theorem.

    An n x n matrix Q is orthogonal if and only if Q^TQ = In (When Q^T is the transpose of Q)


    I'm struggling with this.

    Q = (Q1, Q2, ..., Qn) so Q^T = Row vector of Q.

    So Q^TQ = Is an n x n matrix where the ij components { 1 if i=j or 0 if i =/ j }

    So QTQ = In


    If you can understand what I put, I kind of proved one way. Could someone help with the full proof?
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  2. #2
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    I think you have it.

    For column vectors v_1, v_2, ...,

    Just mention that each entry of the product is (Q^TQ)_{ij} = v_i \cdot v_j, so { 1 if i=j and 0 if i =/ j } by orthogonality.
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  3. #3
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    I am almost positive I have this proven in the Sticky in this forum.

    I could be wrong but I think it is there.
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  4. #4
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    Quote Originally Posted by dwsmith View Post
    I am almost positive I have this proven in the Sticky in this forum.

    I could be wrong but I think it is there.
    I searched but could not find it
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  5. #5
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    Quote Originally Posted by snowtea View Post
    I think you have it.

    For column vectors v_1, v_2, ...,

    Just mention that each entry of the product is (Q^TQ)_{ij} = v_i \cdot v_j, so { 1 if i=j and 0 if i =/ j } by orthogonality.
    I haven't though, as it is an if and only if statement, and I've only shown it one way...
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  6. #6
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    Is anyone able to give me the full proof?
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  7. #7
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    Iff. two ways.

    1 direction
    P\Rightarrow Q

    Sticky PDF #23
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