1. Let w1 be the first principal component as in the previous question. Now, suppose we would like to find a second unit vector, w2 ∈ Rd , that maximizes the variance of _w2,x_, but is also uncorrelated to _w1,x_. That is, we would like to solve argmax w:           w            =1, E[(_w1,x_)(_w,x_)]=0 Var[_w,x_]. Show that the solution to this problem is to set w to be the second principal component of x1, . . .,xm. Hint: Note that E[(_w1,x_)(_w,x_)] = w_ 1 E[xx_]w = mw_ 1 Aw, where A =_ i xix_ i . Since w is an eigenvector of A we have that the constraint

E[(_w1,x_)(_w,x_)] = 0 is equivalent to the constraint

_w1,w_ = 0.