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:
and for that to be true for all possible $x$ and $y$ then we must have:
i.e. the matrix inverse is equal to the transpose.

$x_{T}y=(Ox)_{T}(Oy)=x_{T}O_{T}Oy$

$O_{T}O=I$

These matricese are called the orthogonal matrices.

TODO is there any more intuitive way to think about this?

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