Canonical Correlation

Suppose $A_1$ is an $n\times m_1$ matrix and $A_2$ is an $n\times m_2$ matrix. The cosine of the angle between two linear combinations $A_1x_1$ and $A_2x_2$ is $$f(x_1,x_2)=\frac{x_1'A_1'A_2x_2}{\sqrt{x_1'A_1'A_1x_1}\sqrt{x_2'A_2'A_2x_2}}.$$ Consider the problem of finding the maximum, the minimum, and possible other stationary values of $f$.

are two matrices of dimensions, respectiSpecifically there exists a non-singular $K$ of order $n_1$ and a non-singular $L$ of order $n_2$ such that $$\begin{align*} K'A_1'A_1^{\ }K&=I_1,\\ L'A_2'A_2^{\ }L&=I_2,\\ K'A_1'A_2^{\ }L&=D. \end{align*}$$ Here $I_1$ and $I_2$ are diagonal, with the $n_1$ and $n_2$ leading diagonal elements equal to one and all other elements zero. $D$ is a matrix with the non-zero canonical correlations in non-increasing order along the diagonal and zeroes everywhere else.

http://en.wikipedia.org/wiki/Principal_angles http://meyer.math.ncsu.edu/Meyer/PS_Files/AnglesBetweenCompSubspaces.pdf