= The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
Let's show that this definition is equivalent to <the orthogonal group is the group of all matrices that preserve the dot product>.
Note that:
$$
x^Ty = (Ox)^T (Oy) = x^T O^T O y
$$
and for that to be true for all possible $x$ and $y$ then we must have:
$$
O^T O = I
$$
i.e. the <matrix inverse> is equal to the <transpose>.
Conversely, if:
$$
O^T O = I
$$
is true, then
$$
(Ox)^T (Oy) = x^T (O^T O) y = x^Ty
$$
These matricese are called the <orthogonal matrices>.
TODO is there any more intuitive way to think about this?