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 Wo , that is, assume that Wo is an orthonormal matrix.{b} [10 points] Let Do be a diagonal matrix containing the standard deviations of {X3- : j = 1, ....p} Show that scaling i from part (a) via: yo = Dgli may change the covariance matrix. i.e. E = Full?) 9E Vary-{X} = En. in general. 'Write Vaer") in terms of the elements of En. Considering the structure of D5, what would be the [1%. j)-th element of E? {c} [10 points] Following from part {b}. show that in general the eigenvalues and eigenvectors of B will be different than those of $0. 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 denition of an eigenvectorf eigenvalue pair does not necessarily hold