Let L - D - A denote the unnormalized Laplacian matrix for an undirected, weighted graplh G(V,8), where A is a symmetric, nonnegative adjacency matrix, and D- diag[s1,..., sv] is a diagonal matrix that encodes the nodes, strengths, s,-, Aij > 0 (also called weighted degrees). Let N = |V| and M = denote the number of nodes and edges, respectively. Define L = D-L I-D-1A to be the asymmetric-normalized Laplacian and L - D-1/2LD-1/2- I-D-1/2AD-1/2 to be the symmetric-normalized Laplacian 1. (9 points) Prove the following: (A) Prove xTLx = j Aij (zi-y)2 for any vector x RN. (B) Let w denote a diagonal matrix of size M M having diagonal entries that encode the edge weights: Wee-Aij for edge e-(i, j). Further, let U denote a matrix of size M N in which each entry Uej encodes whether node j is the start or end of edge e: 1 if edge e is (j,.) -1 if edge e is (,j) 0 otherwise Uei-- Prove that L JTWU and find a matrix Q such that L QTQ (C) Use your answer from either problem 1(A) or 1(B) to prove that the eigenvalues of L are real and nonnegative. (Which proves L is a nonnegative-definite matrix.) 1. (9 points) Prove the following: (A) Prove xTLx = 'i A,(zi-1,)2 for any vector x RN (B) Let w denote a diagonal matrix of size M M having diagonal entries that encode the edge weights: Wee-Aij for edge e (i,j). Further, let U denote a matrix of size Mx N in which each entry Uej encodes whether node j is the start or end of edge e: 1 if edge e is (j,) -1 if edge e is (J) 0 otherwise Ue.j Prove that L = UTWU and find a matrix Q such that L = QTQ (C) Use your answer from either problem 1(A) or 1(B) to prove that the eigenvalues of L are real and nonnegative. (Which proves L is a nonnegative-definite matrix.) Let L - D - A denote the unnormalized Laplacian matrix for an undirected, weighted graplh G(V,8), where A is a symmetric, nonnegative adjacency matrix, and D- diag[s1,..., sv] is a diagonal matrix that encodes the nodes, strengths, s,-, Aij > 0 (also called weighted degrees). Let N = |V| and M = denote the number of nodes and edges, respectively. Define L = D-L I-D-1A to be the asymmetric-normalized Laplacian and L - D-1/2LD-1/2- I-D-1/2AD-1/2 to be the symmetric-normalized Laplacian 1. (9 points) Prove the following: (A) Prove xTLx = j Aij (zi-y)2 for any vector x RN. (B) Let w denote a diagonal matrix of size M M having diagonal entries that encode the edge weights: Wee-Aij for edge e-(i, j). Further, let U denote a matrix of size M N in which each entry Uej encodes whether node j is the start or end of edge e: 1 if edge e is (j,.) -1 if edge e is (,j) 0 otherwise Uei-- Prove that L JTWU and find a matrix Q such that L QTQ (C) Use your answer from either problem 1(A) or 1(B) to prove that the eigenvalues of L are real and nonnegative. (Which proves L is a nonnegative-definite matrix.) 1. (9 points) Prove the following: (A) Prove xTLx = 'i A,(zi-1,)2 for any vector x RN (B) Let w denote a diagonal matrix of size M M having diagonal entries that encode the edge weights: Wee-Aij for edge e (i,j). Further, let U denote a matrix of size Mx N in which each entry Uej encodes whether node j is the start or end of edge e: 1 if edge e is (j,) -1 if edge e is (J) 0 otherwise Ue.j Prove that L = UTWU and find a matrix Q such that L = QTQ (C) Use your answer from either problem 1(A) or 1(B) to prove that the eigenvalues of L are real and nonnegative. (Which proves L is a nonnegative-definite matrix.)