16:59 . . I .. . . assignment 2 (202302).pdf . . . Problem 1 Let H be a circulant matrix ho hN-1 ... . . . . . . h2 hi 21 ho . . . . . . . .. h3 h2 H = h1 . . . . . . . . . ha h3 E RNXN (1) : hN-1 hN-2 . . . . . . . . . ho and let When @132 ... " ( N - 1) 1 7 , k = 0, 1, . . ., N - 1. (2) (a) (10%) Show that by is an eigenvector of H. (b) (10%) Show that F = [bo b1 . .. by-1] (3) satisfies FFF = FF# = I. Problem 2 (30%) Let A E Imam be a link matrix of a graph of m nodes, where [A]ji = 1/c; if there is an edge from node i to node j, and node i has a total number of c; outgoing edges. Suppose that the graph is strongly connected, that is, for every pair of node (i, j) in the graph, there exists a path of edges that links node i and node j. So node j can be reached from node i by a finite number of steps. (a) Show that [A2]ji > 0 if and only if node j can be reached by node i in exactly two steps. (b) Show that B = m(I + A + A2 + . .. Am-1) is positive and column-stochastic. (c) Show that dim(Null(A - I)) = 1 [Hint: First show that Null(A - I) C Null(B - I)]. Problem 3 (30%) Let A E Roxn, and suppose that A satisfies dij = -aji, for all i, j. Show the following results. (a) A admits a unitary eigendecomposition A = VAVR, where A is a diagonal matrix whose main diagonals are the eigenvalues of A, and V is a unitary matrix whose ith column is an eigenvector corresponding to the ith eigenvalue of A. (b) The eigenvalues of A are purely imaginary or zero. (c) If n is odd, then A has at least one eigenvalue that equals zero. Problem 4 (10%) Let A, B E Inx", and suppose that either A or B is non-singular. If AB is diagonalizable, show that BA is also diagonalizable