# Canonical correlation coefficients

Maximum values of correlation coefficients between pairs of linear functions

$$U = \alpha _ {1} X _ {1} + \dots + \alpha _ {s} X _ {s} ,\ \ V = \beta _ {1} X _ {s+1} + \dots + \beta _ {t} X _ {s+t}$$

of two sets of random variables $X _ {1} \dots X _ {s}$ and $X _ {s+1} \dots X _ {s+t}$ for which $U$ and $V$ are canonical random variables (see Canonical correlation). The problem of determining the maximum correlation coefficient between $U$ and $V$ under the conditions ${\mathsf E} U = {\mathsf E} V = 0$ and ${\mathsf E} U ^ {2} = {\mathsf E} V ^ {2} = 1$ can be solved using Lagrange multipliers. The canonical correlation coefficients are the roots $\lambda _ {1} \geq \dots \geq \lambda _ {s} > 0$ of the equation

$$\left | \begin{array}{cc} - \lambda \Sigma _ {11} &\Sigma _ {12} \\ \Sigma _ {21} &- \lambda \Sigma _ {22} \\ \end{array} \right | = 0 ,$$

where $\Sigma _ {11}$ and $\Sigma _ {22}$ are the covariance matrices of $X _ {1} \dots X _ {s}$ and $X _ {s+1} \dots X _ {s+t}$, respectively, and $\Sigma _ {12} = \Sigma _ {21} ^ {T}$ is the covariance matrix between the variables of the first and second sets. The $r$- th root of the equation is called the $r$- th canonical correlation coefficient between $X _ {1} \dots X _ {s}$ and $X _ {s+1} \dots X _ {s+t}$. It is equal to the maximum value of the correlation coefficients between the pair of linear functions $U ^ {(r)}$ and $V ^ {(r)}$ of canonical random variables, each of which has variance one and is uncorrelated with the first $r - 1$ pairs of variables $U$ and $V$. The coefficients $\alpha ^ {(r)} = ( \alpha _ {1} ^ {(r)} \dots \alpha _ {s} ^ {(r)} ) ^ {T}$, $\beta ^ {(r)} = ( \beta _ {1} ^ {(r)} \dots \beta _ {t} ^ {(r)} ) ^ {T}$ of $U ^ {(r)}$ and $V ^ {(r)}$ satisfy the equation

$$\left ( \begin{array}{cc} - \lambda \Sigma _ {11} &\Sigma _ {12} \\ \Sigma _ {21} &- \lambda \Sigma _ {22} \\ \end{array} \right ) \left ( \begin{array}{c} \alpha \\ \beta \end{array} \right ) = 0$$

when $\lambda = \lambda _ {r}$.