# Proving that the product of orthogonal matrices is also orthogonal.

• Aug 15th 2011, 04:23 AM
Glitch
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?
• Aug 15th 2011, 04:32 AM
FernandoRevilla
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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