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Thread: 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:

    $\displaystyle \[
    A =
    \begin{array}{cc}
    a & b \\
    c & d \\
    \end{array}
    \]$

    $\displaystyle \[
    B =
    \left( {\begin{array}{cc}
    e & f \\
    g & h \\
    \end{array} } \right)
    \]$

    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:

    $\displaystyle a^2ef + b^2gh = 0$
    $\displaystyle c^2ef + d^2gh = 0$

    The best I could do was show that:

    $\displaystyle (a^2 - c^2)(ef) = (b^2 - d^2)(ef)$

    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.

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