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Math Help - Proving that the product of orthogonal matrices is also orthogonal.

  1. #1
    Senior Member
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    Proving that the product of orthogonal matrices is also orthogonal.

    I'm not sure how to do this. I tried making two general 2 x 2 matricies like so:

    \[<br />
A =<br />
\begin{array}{cc}<br />
 a & b  \\<br />
 c & d  \\<br />
\end{array}<br />
\]

    \[<br />
B =<br />
\left( {\begin{array}{cc}<br />
 e & f  \\<br />
 g & h  \\<br />
 \end{array} } \right)<br />
\]

    To be orthogonal, these equations must be satisfied (via dot product):
    ab + cd = 0
    ef + gh = 0

    Multiplying the two matricies, and then finding the dot product of the result yeilds:

    a^2ef + b^2gh = 0
    c^2ef + d^2gh = 0

    The best I could do was show that:

    (a^2 - c^2)(ef) = (b^2 - d^2)(ef)

    I think I'm attempting this wrong. :/ Any advice?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Proving that the product of orthogonal matrices is also orthogonal.

    M\in\mathbb{R}^{2\times 2} is orthogonal iff M^tM=I . So, if A,B\in\mathbb{R}^{2\times 2} are orthogonal then, (AB)^t(AB)=\ldots=I which implies AB orthogonal.
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