Generalized inverses and their application to applied probability problems

Hunter, Jeffrey J.

doi:10.1016/0024-3795(82)90218-x

Cited by 73 publications

(96 citation statements)

References 6 publications

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“…Regarding the fundamental matrix we draw attention to an important aspect of such map, namely, that Z is a particular example of a generalized inverse of the operator I − P [24,37]. As it is well-known, such inverses are in general nonunique and possess a rich algebraic theory in connection with applications to stochastic processes, linear estimation, difference equations and other areas [48].…”

Section: Motivation and Statement Of The Problems: Hitting Times And mentioning

confidence: 99%

“…">Appendix: proofsThroughout the proofs we recall the notation |e I = |e I n k as n, k are clear from context.Proof of Proposition 6.3. The proof is inspired by [24]. We recall that if M ij denotes the (i, j)-th minor of a matrix A, we define the adjugate matrix as adj(A) := ((−1) i+j M ji ) 1≤i,j≤n , and it holds that A adj(A) = adj(A)A = det(A)I By Sylvester's determinant theorem, we have det(I m + AB) = det(I n + BA), from which we obtain as a consequence that det(X + |c r|) = det(X) + r| adj(X)|c…”

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confidence: 99%

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Mean hitting times of quantum Markov chains in terms of generalized inverses

Lardizabal

2019

Quantum Inf Process

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We study quantum Markov chains on graphs, described by completely positive maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008) and which includes the dynamics given by open quantum random walks as defined by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such structures we examine a quantum notion of mean time of first visit to a chosen vertex. However, instead of making direct use of the definition as it is usually done, we focus on expressions for such quantity in terms of generalized inverses associated with the walk and most particularly the socalled fundamental matrix. Such objects are in close analogy with the theory of Markov chains and the methods described here allow us to calculate examples that illustrate similarities and differences between the quantum and classical settings. arXiv:1907.01313v2 [math-ph] 9 Jul 2019 Appendix: proofsThroughout the proofs we recall the notation |e I = |e I n k as n, k are clear from context.Proof of Proposition 6.3. The proof is inspired by [24]. We recall that if M ij denotes the (i, j)-th minor of a matrix A, we define the adjugate matrix as adj(A) := ((−1) i+j M ji ) 1≤i,j≤n , and it holds that A adj(A) = adj(A)A = det(A)I By Sylvester's determinant theorem, we have det(I m + AB) = det(I n + BA), from which we obtain as a consequence that det(X + |c r|) = det(X) + r| adj(X)|c

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Section: Motivation and Statement Of The Problems: Hitting Times And mentioning

confidence: 99%

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confidence: 99%

Mean hitting times of quantum Markov chains in terms of generalized inverses

Lardizabal

2019

Quantum Inf Process

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“…where and are given by (3) and (2) respectively, is defined in (4) and in (5). Finally, it is worthy to note that the matrix A # = Z − Π is the group generalized inverse [6] of the matrix (I − P ) and may be used in the same manner as Z (other generalized inverses [3] may also be used).…”

Section: Comparison Of Markov Chainsmentioning

confidence: 99%

“…Finally, it is worthy to note that the matrix A # = Z − Π is the group generalized inverse [6] of the matrix (I − P ) and may be used in the same manner as Z (other generalized inverses [3] may also be used). Furthermore, similar perturbation bounds of those cited in [2,8] may be derived in the same manner as in the classical results of perturbation theory.…”

Section: Comparison Of Markov Chainsmentioning

confidence: 99%

Perturbation results for comparison of Markov models

Rabta¹

2013

J. Stat. Appl. Pro.

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Markov chains are useful to model various complex systems. In numerous situations, the underlying Markov chain is subject to changes. For example, states may be added or deleted and transition probabilities perturbed. It is therefore, necessary to ensure the robustness of the system and to estimate the resulting deviation in the characteristics. In this paper we study the sensitivity of finite Markov chains subject to changes in their state space and propose updating formulas and perturbation bounds.

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“…Actually, since we also have 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].…”

Section: Preliminariesmentioning

confidence: 99%

The Deviation Matrix of a Continuous-Time Markov Chain

Coolen‐Schrijner

Doorn

2002

Prob. Eng. Inf. Sci.

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Abstract. The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P (.) and ergodic matrix Π is the matrixWe give conditions for D to exist and discuss properties and a representation of D. The deviation matrix of a birth-death process is investigated in detail. We also describe a new application of deviation matrices by showing that a measure for the convergence to stationarity of a stochastically increasing Markov chain can be expressed in terms of the elements of the deviation matrix of the chain.

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Generalized inverses and their application to applied probability problems

Cited by 73 publications

References 6 publications

Mean hitting times of quantum Markov chains in terms of generalized inverses

Mean hitting times of quantum Markov chains in terms of generalized inverses

Perturbation results for comparison of Markov models

The Deviation Matrix of a Continuous-Time Markov Chain

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