Editing Canonical correlation (section)

===Solution===
The solution is therefore:
* <math>c</math> is an eigenvector of <math>\Sigma_{XX}^{-1/2} \Sigma_{XY} \Sigma_{YY}^{-1} \Sigma_{YX} \Sigma_{XX}^{-1/2}</math>
* <math>d</math> is proportional to <math>\Sigma _{YY}^{-1/2} \Sigma_{YX} \Sigma_{XX}^{-1/2} c</math>

Reciprocally, there is also:
* <math>d</math> is an eigenvector of <math>\Sigma_{YY}^{-1/2} \Sigma_{YX} \Sigma_{XX}^{-1} \Sigma_{XY} \Sigma_{YY}^{-1/2}</math>
* <math>c</math> is proportional to <math>\Sigma_{XX}^{-1/2} \Sigma_{XY} \Sigma_{YY}^{-1/2} d</math>

Reversing the change of coordinates, we have that
* <math>a</math> is an eigenvector of <math>\Sigma_{XX}^{-1} \Sigma_{XY} \Sigma_{YY}^{-1} \Sigma_{YX}</math>,
* <math>b</math> is proportional to <math>\Sigma_{YY}^{-1} \Sigma_{YX} a;</math>
* <math>b</math> is an eigenvector of <math>\Sigma _{YY}^{-1} \Sigma_{YX} \Sigma_{XX}^{-1} \Sigma_{XY},</math>
* <math>a</math> is proportional to <math>\Sigma_{XX}^{-1} \Sigma_{XY} b</math>.

The canonical variables are defined by:

:<math>U = c^T \Sigma_{XX}^{-1/2} X = a^T X</math>

:<math>V = d^T \Sigma_{YY}^{-1/2} Y = b^T Y</math>