统计学习精要 (Elements of Statistical Learning ) 习题 3.20

统计学习精要 (Elements of Statistical Learning ) 习题 3.20

问题：Consider the canonical-correlation problem (3.67). Show that the leading pair of canonical variates $u_1$ and $v_1$ solve the problem

$$\underset{u^T(Y^T Y )u = 1\\ v^T (X^T X)v =1 }\max u^T(Y^T X )v,$$
a generalized SVD problem. Show that the solution is given by $u_1 = (Y^T Y)^{-\frac{1}{2}} u_1^*$, and $v_1 = (X^T X)^{-\frac{1}{2}} v_1^*$, where $u_1^*$ and $v_1^*$ are the leading left and right singluar vectors in

(YTY)−12(YTX)(