# Statistics: Canonical Correlation

### From Wiki1

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
Σ_{12}, 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: