x= [ and D [ 0.8 Let n be the size of data and d= 2 be the dimension of data. Let X e Rnx2 be the augmented data matrix. You are given the matrix 2.36 -0.48 XTX= , which has eigen-decomposition PDPT, -0.48 2.64 [0.8 -0.6 2 0 where P= 0.6 03 (a) Calculate the upper left entry of PD; (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 "X1,1 + 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 ||1||=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)? x= [ and D [ 0.8 Let n be the size of data and d= 2 be the dimension of data. Let X e Rnx2 be the augmented data matrix. You are given the matrix 2.36 -0.48 XTX= , which has eigen-decomposition PDPT, -0.48 2.64 [0.8 -0.6 2 0 where P= 0.6 03 (a) Calculate the upper left entry of PD; (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 "X1,1 + 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 ||1||=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)