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?


Discussion on "The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose" by Ciro Santilli (@cirosantilli, 34)