What is orthogonal transformation? When $A^{T}A = AA^{T} = I$, I am told it is an orthogonal transformation $A$. But don't really get what it means. Hope to hear some explanations.
$\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix} \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$

The equation $A^T A = A A^T$ says that $A^{-1} = A^T$ so, at least, orthogonal matrices are easy to invert. What does it mean? Matrices represents linear transformation (when a basis is given). 
Orthogonal matrices represent transformations that preserves length of vectors and all angles between vectors, and all transformations that preserve length and angles are orthogonal.    Examples are rotations (about the origin) and reflections in some subspace. (and those are really all examples)
