3. Suppose you have the following covariance matrix: -3 2- (67) = -3 4 (a) (A) Calculate the singular value decomposition of , showing all steps. (b) (A) Using part (a), calculate -. You do not need to simplify your result. (c) (A) Using part (a), what is the first principal component of ? What percent of the variance is explained by this first principal component? (d) (C) Verify your work in parts (a) - (c) by calculating the same results numeri- cally. (e) Now, suppose X ~ N(0,): i. (A) What is the distribution of XTV(1), where v(1) is the first principal component from part (c)? ii. (C) Simulate 1000 random vectors from X ~ N2(0,). Using these random vectors, plot a histogram of XTV (1) and overlay the theoretical density on top of this histogram.