Let n be the size of data and d = 2 be the dimension of data. Let X ERnx2 be the augmented data matrix. You are given the matrix 2.36 -0.48 0.8 -0.6 2 0 n-1 XT X = which has eigen-decomposition PDPT, where P = and D= -0.48 2.64 0.6 0.8 0 [. :) (a) Calculate the upper left entry of PD; n 1 (b) Under what assumptions on X does it hold that n-1 XT X is the sample covariance matrix? (Make sure you give a precise mathematical statement, i.e. write something like "X;,1 + X1,2 = 1 for all i = 1, 2, ...,, n" and do NOT write ambiguous statements like "Data are centered". ) (c) What are possible values of n? (If you believe all positive integers are possible, you can answer "1, 2, ...") (d) Find arg max|v||=1 || Xv||2 (e) What is the first principle component such that the projecting data maximizes variance? (f) What is the magnitude of projection of [1 1]to the the principal component found in (e)