Statistics: Canonical Correlation
Let a random vector with . Let and
There are many ways of arriving at the canonical correlations between and . Here we present an approach based on the singular value decomposition of an arbitrary matrix.
Let where and are orthogonal matrices and is a diagonal (not necessarily square) matrix, with dimension the same as that of Σ12, and with non-negative diagonal elements ordered from largest to smallest.
Then, let and
so that and
Many of the major matrices of linear models can be expressed in terms of canonical correlations: