12.4. (Sec. 12.2) Let XO) =AZ + yO), X(2) = BZ + y(2),

where y(1), y(2), Z are independent with mean zero and covariance matrices I with appropriate dimensionalities. Let A = (a l , ... , ak)' B = (b l , •.. , bk), and suppose that A' A, B' B are diagonal with positive diagonal elements. Show that the canonical variables for nonzero canonical correlations are proportional to a~X{l), b;X(2). Obtain the canonical correlation coefficients and appropriate normalizing coefficients for the canonical variables.