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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|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

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

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

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

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

====Comments====
See also [[Correlation|Correlation]]; [[Correlation coefficient|Correlation coefficient]].