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Math Help - Orthogonal matrix

  1. #1
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    Orthogonal matrix

    I need to prove/disprove that if A^2 is orthogonal matrix, than A is also orthogonal.

    I think the statement is true, because if A^2 orthogonal, then A^tA^t = A^-1A^-1, hence AAA^tA^t = AAA^-1A^-1 = AIA^-1 = AA^-1 = I, hence A is orthogonal.

    Is my proof right?
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  2. #2
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    Re: Orthogonal matrix

    Quote Originally Posted by rebecca View Post
    I need to prove/disprove that if A^2 is orthogonal matrix, than A is also orthogonal.

    I think the statement is true, because if A^2 orthogonal, then A^tA^t = A^-1A^-1, hence AAA^tA^t = AAA^-1A^-1 = AIA^-1 = AA^-1 = I, hence A is orthogonal.

    Is my proof right?
    Your first statement doesn't necessarily follow.

    We know that

    $(AA)^T(AA)=(AA)(AA)^T=I$

    and

    $(AA)^T=A^TA^T$

    so

    $(AA)^{-1}=(A^TA^T)$

    but we don't know that $A^T=A^{-1}$ yet
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  3. #3
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    Re: Orthogonal matrix

    You're going to have a hard time proving this, because it's not true.

    Consider the matrix:

    $A = \begin{bmatrix}1&2\\-1&-1\end{bmatrix}$

    Now $A^2 = -I$ and $-I$ is an orthogonal matrix:

    $(-I)^T = -I$ and $(-I)(-I)^T = (-I)(-I) = I$

    However, $A$ is not orthogonal.
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  4. #4
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    Re: Orthogonal matrix

    (aa)(aa)'=i
    (aa)(a'a')=i
    a[(aa')a']=i
    (aa')a'=a'
    aa'=i
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  5. #5
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    Re: Orthogonal matrix

    Quote Originally Posted by Hartlw View Post
    (aa)(aa)'=i
    (aa)(a'a')=i
    a[(aa')a']=i


    (aa')a'=a'


    how do you justify this step?

    a'a != I
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  6. #6
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    Re: Orthogonal matrix

    Quote Originally Posted by romsek View Post
    [/B]

    how do you justify this step?

    a'a != I
    I don't.
    (aa')a'=a-1
    You're right. You don't know a'=a-1
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  7. #7
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    Re: Orthogonal matrix

    (AB)-1=B-1A-1
    A'A'=A-1A-1
    if A2=B2 CAN YOU SHOW A=+-B ? I can't.
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  8. #8
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    Re: Orthogonal matrix

    Quote Originally Posted by Hartlw View Post
    (AB)-1=B-1A-1
    A'A'=A-1A-1
    if A2=B2 CAN YOU SHOW A=+-B ? I can't.
    Deveno already gave a counter example. The original assertion isn't true.
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  9. #9
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    Re: Orthogonal matrix

    Quote Originally Posted by romsek View Post
    Deveno already gave a counter example. The original assertion isn't true.
    That doesn't answer my question:


    if A2=B2 CAN YOU SHOW A=+-B ? If you can, A'=+-A-1, which is an interesting result. Guess I haven't struck an intellectual curiousity chord.
    I'll have to get the answer myself.
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  10. #10
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    Re: Orthogonal matrix

    Quote Originally Posted by Hartlw View Post
    That doesn't answer my question:


    if A2=B2 CAN YOU SHOW A=+-B ? If you can, A'=+-A-1, which is an interesting result. Guess I haven't struck an intellectual curiousity chord.
    I'll have to get the answer myself.
    I admit I didn't read closely enough to see that you were asking a general question. I will take a look.
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  11. #11
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    Re: Orthogonal matrix

    Does X2=A2 → X=ĪA?
    No, because matrices have divisors of 0. Explanation:
    X2-A2=(X-A)(X+A)=0 does not imply X-A=0 or X+A=0 for matrices.

    If you prefer a counter example, that allegedly has been done for a 2X2 matrix (I tend not to work through Devenoís posts, but Iíll take romseksí word for it).

    OK, Iím happy. My question is answered. Since everyone else was apparently happy before my posts, we are all happy now.

    EDIT: In answer to the OP, one could say if A2 is orthogonal, A could be orthogonal, but not necessarily.
    Last edited by Hartlw; May 30th 2014 at 01:05 PM.
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