Ciro Santilli @cirosantilli 35

When we have a symmetric matrix, a change of basis keeps symmetry iff it is done by an orthogonal matrix, in which case:

The general result from eigendecomposition of a matrix:becomes:where $O$ is an orthogonal matrix, and therefore has $O^{-1}=O^T$.

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?

Complex analogue of orthogonal matrix.

Applications: