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Math Help - Orthogonal Matrix problems

  1. #1
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    Orthogonal Matrix problems

    This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.

    Let A and B be orthogonal n\times n matrices. Determine which of the following are orthogonal (provide reasons for answers).

    i) A^{-1} (my answer is below, but I'm not sure if it is concrete enough)
    Let A=[a_{jk}]. Since A is orthogonal, then A^{-1}=A^T=[a_{kj}]. Therefore, A^{-1} is orthogonal.

    ii) A-B
    (I'm afraid I'm not sure about this one)

    If AC is also an orthogonal n\times n matrix, must C be orthogonal?

    I've inferred that C is orthogonal, using the following three equivalent statements for some n\times n matrix Q:
    a. Q is orthogonal.
    b. ||Qx||=||x|| for every x\in R^n
    c. Qx\cdot Qy=x\cdot y for every x,y\in R^n

    The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).

    Can anyone help me with this?
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  2. #2
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    Quote Originally Posted by Runty View Post
    This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.

    Let A and B be orthogonal n\times n matrices. Determine which of the following are orthogonal (provide reasons for answers).

    i) A^{-1} (my answer is below, but I'm not sure if it is concrete enough)
    Let A=[a_{jk}]. Since A is orthogonal, then A^{-1}=A^T=[a_{kj}]. Therefore, A^{-1} is orthogonal.

    ii) A-B
    (I'm afraid I'm not sure about this one)

    If AC is also an orthogonal n\times n matrix, must C be orthogonal?

    I've inferred that C is orthogonal, using the following three equivalent statements for some n\times n matrix Q:
    a. Q is orthogonal.
    b. ||Qx||=||x|| for every x\in R^n
    c. Qx\cdot Qy=x\cdot y for every x,y\in R^n

    The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).

    Can anyone help me with this?
    i) I=AA^T=(A^T)^T A^T. so A^T=A^{-1} is orthogonal.

    ii) not necessarly. for example we might have A=B.

    iii) yes because we have CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    i) I=AA^T=(A^T)^T A^T. so A^T=A^{-1} is orthogonal.

    ii) not necessarly. for example we might have A=B.

    iii) yes because we have CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.
    Thanks for those answers. Are you absolutely sure about the first one? It seems a little... incomplete to me. Not wrong, but a tad lacking in substance.
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