Q1) Let X = (X1, ..., Xp) be a p x 1 random vector and u be a p x 1 vector of real-valued constants such that E(X) = M. Let Var(X) = Zo, i.e. Zo is the covariance matrix for X. Assume that Zo has an eigenvalue decomposition and let the expression for the normed eigenvalue decomposition of Zo be: Eo = Wodo WOT, that is, assume that Wo is an orthonormal matrix. (a) Show that centering X via: X =X-/ does not change the covariance matrix, i.e. Var(X) = Var(X). (b) Let D. be a diagonal matrix containing the standard deviations of {Xj : j = 1, ...p}. Show that scaling X from part (a) via: X* = D' x may change the covariance matrix, i.e. E = Var(X*) # Var(X) = Zo in general. Write Var(X*) in terms of the elements of Zo. Considering the structure of D., what would be the (i, j)-th element of E? (c) Following from part (b), show that in general the eigenvalues and eigenvectors of E will be different than those of Zo. This shows in particular that scaling the data will almost always result in different principal components. Hint: Use an arbitrary eigenvector to show that the definition of an eigenvector/eigenvalue pair does not necessarily hold. (d) Recall that E = Var(X*). Assume that E has the following eigenvalue decomposi- tion: E = WAWT. and that this decomposition has unique and non-zero eigenvalues and All > Azz > ... > App. Let V = QW1 X* + BW2 X* where W; is the j-th column of W for (a, 3) E R2. In other words, V is a linear combination of the first two principal component scores from E for a vector X*. Derive the expected value and variance of V