Ciro Santilli @cirosantilli 34

The transpose and matrix inverse commute:

A matrix that equals its transpose:

Can represent a symmetric bilinear form as shown at matrix representation of a symmetric bilinear form, or a quadratic form.

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.

Conversely, if:is true, then

These matricese are called the orthogonal matrices.

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