If A and B are both orthogonal nxn matrices.
Can anyone show That AB is orthogonal...
Thanks
An $\displaystyle n\times n$ $\displaystyle A$ is orthogonal iffOriginally Posted by dopi
$\displaystyle AA^T=I$
Now suppose $\displaystyle A$ and $\displaystyle B$ be orthogonal, and let
$\displaystyle C=AB$. Now $\displaystyle C^T=B^TA^T$, so:
$\displaystyle
CC^T=ABB^TA^T=AIA^T=AA^T=I
$
Hence $\displaystyle C$ is orthogonal.
RonL
From the definition, we have for A and B:
$\displaystyle
\begin{array}{l}
AA^{ - 1} = A^{ - 1} A = I \\
BB^{ - 1} = B^{ - 1} B = I \\
\end{array}
$
$\displaystyle
\left( {AB} \right)\left( {AB} \right)^{ - 1} = ABB^{ - 1} A^{ - 1} = AIA^{ - 1} = AA^{ - 1} = I
$