Canonical Correlation Clarified by Singular Value Decomposition

You want to find a linear combination of the x coordinates that correlates well over the data with an (in general, different) linear combination of the y coordinates. In fact, you want to find the best such matched pair of linear combinations on the x and y sides, that is, the one yielding the largest coefficient of correlation. But why stop there? Once you have the best pair, you can ask for the second-best pair, that is, the linear combination of x coordinates in the subspace orthogonal to the first combination (on the x side) that best correlates with a linear combination of y coordinates in the subspace orthogonal to the first combination (on the y side). And so on for further pairs or linear combinations in remaining orthogonal subspaces. We can proceed like this until we have exhaused the orthogonal subspaces on either the x or the y side, whichever comes first. So the number of pairs (termed the number of canonical coordinates) is d = min[rank(x), rank(y)], where rank means column rank. We denote the matrices of coefficients of the linear combinations by a (on the x side), of size p1 × d, and b (on the y side), of size p2 × d. If matrices u and v denote the linear combinations evaluated for each data point, we have