On the spectra of nonsymmetric Laplacian matrices

Abstract

A Laplacian matrix is a square real matrix with nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with the absolute values of the off-diagonal entries not exceeding 1/n, where n is the order of the matrix. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of a standardized Laplacian matrix are equal to the in-forest dimension of the corresponding digraph and one less than the in-forest dimension of the complementary digraph, respectively. These eigenvalues are semisimple. The spectrum of a standardized Laplacian matrix belongs to the meet of two closed disks, one centered at 1/n, another at 1-1/n, each having radius 1-1/n, and two closed angles, one bounded with two half-lines drawn from 1, another with two half-lines drawn from 0 through certain points. The imaginary parts of the eigenvalues are bounded from above by 1/(2n) cot(pi/2n); this maximum converges to 1/pi as n goes to infinity. Keywords: Laplacian matrix; Laplacian spectrum of graph; Weighted directed graph; Forest dimension of digraph; Stochastic matrixComment: 11 page

Learning Representation and Control in Markov Decision Processes: New Frontiers

“…On graphs, the discretized (symmetric and nonsymmetric) Laplacian has been studied extensively in graph theory, where its spectra reveal structural properties of undirected and directed graphs [26,27]. Stated in its most general form, the (nonsymmetric) Laplacian matrix is one whose off-diagonal elements are nonpositive and whose row sums are equal to 0 [2,22,24]. As we show in this paper, there are strong connections between Laplacian matrices and MDPs.…”

“…Virtually everything that one wants to know about a chain can be extracted from A and its Drazin inverse. We denote transition matrices generally as P , and define the Laplacian associated with a transition matrix as L = I − P [2,22,24] (see Figure 1.3). It has long been known that the Drazin inverse of the singular Laplacian matrix L reveals a great deal of information about the structure of the Markov chain [92,121].…”

“…Also, the sensitivity of the invariant distribution of an ergodic Markov chain to perturbations in the transition matrix can be quantified by the size of the entries in the Drazin inverse of the Laplacian. 2 The solution to average-reward and discounted MDPs can be shown to depend on the Drazin inverse of the Laplacian [21,110].…”

“…It can be shown that the operator T * is a contraction in a Hilbert space, and hence the algorithm will asymptotically converge [138]. 2 A similar procedure can be used to compute the optimal action value function Q * (s, a). A drawback of value iteration is that its convergence can be slow for γ near 1.…”

“…Definition 3.1. The generalized Laplacian operator [2,22,24] is defined as a (possibly nonsymmetric) matrix L ∈ R n×n , where…”