Editing Canonical correlation (section)

==Connection to principal angles==
Assuming that <math>X = (x_1, \dots, x_n)^T</math> and <math>Y = (y_1, \dots, y_m)^T</math> have zero [[expected value]]s, i.e., <math>\operatorname{E}(X)=\operatorname{E}(Y)=0</math>, their [[covariance]]  matrices <math>\Sigma _{XX} =\operatorname{Cov}(X,X) = \operatorname{E}[X X^T]</math> and <math>\Sigma _{YY} =\operatorname{Cov}(Y,Y) = \operatorname{E}[Y Y^T]</math> can be viewed as [[Gram matrix|Gram matrices]] in an [[inner product]] for the entries of  <math>X</math> and <math>Y</math>, correspondingly. In this interpretation, the random variables, entries <math>x_i</math> of  <math>X</math> and <math>y_j</math> of <math>Y</math> are treated as elements of a vector space with an inner product given by the [[covariance]] <math>\operatorname{cov}(x_i, y_j)</math>; see [[Covariance#Relationship to inner products]].

The definition of the canonical variables <math>U</math> and <math>V</math> is then equivalent to the definition of [[principal angles|principal vectors]] for the pair of subspaces spanned by the entries of  <math>X</math> and <math>Y</math> with respect to this  [[inner product]]. The canonical correlations <math>\operatorname{corr}(U,V)</math> is equal to the [[cosine]] of [[principal angles]].