Given two column vectors *X* = (*X*_{1}, ..., *X*_{n})′ and *Y* = (*Y*_{1}, ..., *Y*_{m})′ of random variables with finite second moments, one may define the cross-covariance cov(*X*, *Y*) to be the *n*×*m* matrix whose *ij* entry is the covariance cov(*X*_{i}, *Y*_{j}). (Sometimes this is called simply the covariance between *X* and *Y*. But sometimes one speaks of the "covariance" of *X*, intending the *n*×*n* matrix of covariances between the pairs of scalar components of *X*. Sometimes the latter matrix is called the variance of *X*.)

Canonical correlation analysis seeks vectors ** a** and