@MISC{Department02themulticut, author = {Marina Meil Department}, title = {The Multicut Lemma}, year = {2002} }

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Abstract

of P and by the corresponding eigenvectors. Denote by 0 = 1 2 : : : n the eigenvalues of L and by u the corresponding eigenvectors. Then, i = 1 i (5) u (6) for all i = 1; : : : n. Note that this lemma ensures that the eigenvalues of P are always real and the eigenvectors lineraly independent. Lemma 4 (Lumpability) Let P be a matrix with rows and columns indexed by V that has independent eigenvectors. Let = (C 1 ; C 2 ; : : : C k ) be a partition of V . Then, P has K eigenvectors that are piecewise constant w.r.t. and correspond to non-zero eigenvalues if and only if the sums P ik = P ij are constant for all i 2 C l and all k; l = 1; : : : k and the matrix ^ P = [ P kl ] k;l=1;:::K (with ^ P kl = j2C k P ij ; i 2 C l ) is non-singular. Lemma 5 (Relationship between P and ^ P ) Assume that the conditions of Lemma 4 hold. Let v and 1 = 1 2 : : : K be the piecewise constant eigenvectors of P and their eigenvalues. Denote by 1 = ^