Fun with Orthogonal Transformations

An orthogonal transformation is defined well at Wolfram's Math site. Essentially it transforms one Cartesian basis to another.

Consider the constraints placed by the Kronecker delta:

$\delta$ij=ei·ej=[∑(ei·ek')ek']·ej=∑(ei·ek')(ek'·ej) over k

Define
$a$ij≡ei'·ej


$\delta$ij=∑(ei·ek')(ek'·ej)=∑akiakj over k

There is of course a similar set of relations:
$\delta$ij=∑aikajk over k

They place 0.5n(n+1) constraints over n-dimensional transformations, leaving 0.5n(n-1) degrees of freedom.