# Canonical correlation coefficients

Maximum values of correlation coefficients between pairs of linear functions

of two sets of random variables and for which and are canonical random variables (see Canonical correlation). The problem of determining the maximum correlation coefficient between and under the conditions and can be solved using Lagrange multipliers. The canonical correlation coefficients are the roots of the equation

where and are the covariance matrices of and , respectively, and is the covariance matrix between the variables of the first and second sets. The -th root of the equation is called the -th canonical correlation coefficient between and . It is equal to the maximum value of the correlation coefficients between the pair of linear functions and of canonical random variables, each of which has variance one and is uncorrelated with the first pairs of variables and . The coefficients , of and satisfy the equation

when .

#### Comments

See also Correlation; Correlation coefficient.

**How to Cite This Entry:**

Canonical correlation coefficients. I.O. Sarmanov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Canonical_correlation_coefficients&oldid=12606