Meyer C. D. Jr. (1975). The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17: 443-464.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....however, the techniques used to develop the known bounds are extremely useful to us. Let I be the n Theta n identity matrix, and let A = I Gamma P . The group inverse A ] of A is the matrix satisfying AA ] A = A; A ] AA ] A ] AA ] A ] A: Let W = I Gamma AA ] Fact 1 ([11], Theorem 2.3) If P is ergodic, as in the case of the PageRank Markov chain) then W is the n Theta n matrix in which every row is the row vector T . Fact 2 ( 11] Theorem 2.4) If P is the transition matrix of a regular Markov chain 3 , then A ] 1 X k=0 (P k Gamma W ) 4) 3 A ....

....A ] of A is the matrix satisfying AA ] A = A; A ] AA ] A ] AA ] A ] A: Let W = I Gamma AA ] Fact 1 ( 11] Theorem 2.3) If P is ergodic, as in the case of the PageRank Markov chain) then W is the n Theta n matrix in which every row is the row vector T . Fact 2 ([11], Theorem 2.4) If P is the transition matrix of a regular Markov chain 3 , then A ] 1 X k=0 (P k Gamma W ) 4) 3 A Markov chain is regular if for some integer power k 1 of its transition matrix T , every entry in T is positive. In the case of the PageRank matrix this holds for k = ....

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C. D. Meyer, Jr. The role of the group generalized inverse in the theory of finite Markov Chains. SIAM Review, 17(3):443--464, 1975.

....of Q, which, by definition, is the matrix solving the equations ( Q)X = X( Q) X( Q)X = X and ( Q)X( Q) Q. Still assuming S finite, the solution to this system, if it exists, is unique. For more information on generalized inverses and their application to Markov chains we refer to Meyer [19], Hunter [8] and Lamond and Puterman [18] The existence of D when S is infinite will be our concern in Section 4, and in Section 5 we investigate whether relations such as (3.8) and (3.9) remain valid in an infinite setting. A key role in the analysis will be played by the quantities T ij , the ....

C.D. Meyer (1975). The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17, 443-464.

...., and that an ILU factorization A L U 3 implicitly yields such an approximation: L U) Gamma1 A ] As we will see, this interpretation is not entirely correct and is somewhat misleading. The group inverse (see [12] is only one of many possible generalized inverses. It is well known [21] that the group inverse plays an important role in the modern theory of finite Markov chains. However, it is seldom used as a computational tool, in part because its computation requires knowledge of the stationary distribution vector . As it turns out, different preconditioners result ....

C. D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev., 17 (1975), pp. 443--464.

....to abbreviate number of visits to j before time S as N j (S) which forces the reader to decode formulas. Kemeny and Snell [17] call Z Pi the fundamental matrix, and use (E i T j ) rather than (E i T j ) as the matrix of mean hitting times. Our set up seems a little smoother cf. Meyer [13] who calls Z the group inverse of I Gamma P. The name random target lemma for Corollary 14 was coined by Lov asz and Winkler [19] the result itself is classical ( 17] Theorem 4.4.10) Open Problem 34 Portmanteau theorem for occupation times. Can the results of section 2.2 be formulated as a ....

C.D. Meyer Jr. The role of the group generalized inverse in the theory of finite Markov chains. SIAM Review, 17:443--464, 1975.

..... Unfortunately in the context of stochastic automata networks where the infinitesimal generator is in the form of a sum of tensor products, this approach is not possible. We need to adopt a different strategy. From equation (12) we have I Gamma P = Gamma DeltatQ. Furthermore, it is shown in [9] that (I Gamma P ) # = 1 X k=0 (P k Gamma L) 19) where the superscript # denotes the generalized inverse and L is an n Theta n matrix whose rows are all equal to the stationary probability vector . Hence, an approximation to the (generalized) inverse of I Gamma P may be obtained by ....

C.D. Meyer. The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains. Siam Review, Vol. 17, No. 3, July 1975.

....now yield (3.1) A similar calculation shows that (3.1) holds also for the case i = n. From (3. 1) it readily follows that the entries in the first column of Q # are strictly decreasing from first to last and that the first entry is always positive (which is a general result of Meyer [16] for all diagonal entries of the group inverse of a singular and irreducible M matrix and it is also an outcome of Cohen s results described in the introduction) while the last entry is always negative. Next, that Q # 1;1 jQ # n;1 j follows at once from (3.1) To complete the proof we need ....

C. D. Meyer, Jr. The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev., 17:443--464, 1975.

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Carl D. Meyer. The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev., 17:443--464, 1975.

....= # T (I EZ) 1 , and # T e # T = e # T EZ. 3.1) 3.1) The equation (3.1) by Schweitzer [17] gives the first perturbation bound: ## e ## 1 # #Z## #E## . We define # 1 # #Z## . 3.2. Meyer 1980. The second matrix related to C is the group inverse of A. In his papers [11, 12], Meyer showed that the group inverse A # can be used in a similar way Z is used, and, since Z = A # e# T , all relevant information is contained in A # , and the term e# T is redundant (Meyer [14] In fact, in the place of (3.1) we have e # T = # T (I EA # ) 1 , and ....

C. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev., 17 (1975), pp. 443--464.

....Science Foundation grant CCR 9102853. # Mathematics Department, North Carolina State University, Raleigh, North Carolina 27695 8205 (meyer math.ncsu.edu) The work of this author was supported in part by National Science Foundation grants DMS 9020915 and DDM 8906248. 1062 i. c. f. ipsen and c. d. meyer These norm based bounds are not satisfying for two reasons. First, there exist irreducible chains for which the bounds are not tight, so the condition number # may seriously overestimate the sensitivity to perturbations. Secondly, the bounds generally provide little information about the relative ....

....P denotes the transition probability matrix of an n state irreducible Markov chain with stationary distribution # T whose entries satisfy # i 0 and P i # i = 1. We define A = I P and A # denotes the group inverse of A, properties of which can be found in Campbell and Meyer (1991) Meyer (1975), and Meyer (1982) The matrix P = P E is a perturbation of P that represents the transition matrix of another irreducible chain with stationary distribution # T . The perturbation matrix E is not necessarily constrained to be small. We use E (j) to denote the matrix obtained by deleting ....

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C. D. Meyer (1975), The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev., 17, pp. 443--464.

...., of A, which can be characterized as the unique matrix satisfying the three equations AA # A = A, A # AA # = A # , and AA # = A # A. 2. 4) General properties of the group inverse and applications to the theory of finite Markov chains are well documented see [1] 2] 3] 4] 12] [13], 16] 18] Some special properties of A # that are needed to prove this theorem are summarized below. Observe that if A and # T are respectively partitioned as A = I P = A n c d T a nn and # T = # T , # n ) then the group inverse of A is given by A # = I e# ....

C. D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev., 17 (1975), pp. 443--464.

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Meyer C. D. Jr. (1975). The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17: 443-464.

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Meyer C.D. 'The role of the group generalized inverse in the theory of finite Markov chains', SIAM Rev., 17, 443-464, (1975).

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C. D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Review, 17(1975) 443-464.

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C.D. Meyer. The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains. Siam Review, Vol. 17, No. 3, July 1975.

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C.D. Meyer. The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains. Siam Review, Vol. 17, No. 3, July 1975.

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