On the matrix equation Ax = λBx

Erdelyi, Ivan

doi:10.1016/0022-247x(67)90169-2

Cited by 64 publications

(16 citation statements)

References 9 publications

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“…irreducible and aperiodic) Markov chain X on a finite state space X with transition matrix P = (P i,j ) i,j∈X and stationary distribution π = (π i ) i∈X , which is considered to be a row vector of size |X |. Writing Π to be the matrix where each row is π, the fundamental matrix F = (F i,j ) i,j∈X associated with the Markov chain X, first proposed in the work of Kemeny and Snell (1976), is given by…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On resistance distance of Markov chain and its sum rules

Choi

2019

Linear Algebra and its Applications

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Motivated by the notion of resistance distance on graph, we define a new resistance distance between two states on a given finite ergodic Markov chain based on its fundamental matrix. We prove a few equivalent formulations and discuss its relation with other parameters of the Markov chain such as its group inverse, stationary distribution, eigenvalues or hitting time. In addition, building upon existing sum rules for the hitting time of Markov chain, we give sum rules of this new resistance distance of Markov chains that resembles the sum rules of the resistance distance on graph. This yields Markov chain counterparts of various classical formulae such as Foster's first formula or the Kirchhoff index formulae.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On resistance distance of Markov chain and its sum rules

Choi

2019

Linear Algebra and its Applications

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“…In other words, if there is a solution to the equations: AXA = A , XAX= X, and AX = XA. This solution is necessarily unique and is called the group inverse A # of A [2].…”

Section: Introductionmentioning

confidence: 99%

Matrices for whichA^∗andA^†commute

Hartwig¹,

Spindelböck²

1983

Linear and Multilinear Algebra

156

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“…[3], [4]). Название групповая обратная было введено И. Эрдели в [5] в связи с тем, что условия, определяющие эту обобщенную обратную, яв-ляются необходимыми и достаточными условиями для вложимости матрицы A в некоторую мультипликативную группу. А именно, множество всех степеней матрицы A и всех степеней матрицы A является циклической группой относи-тельно обычного матричного умножения, нейтральным элементом этой группы является матрица AA , обратным элементом к A k является элемент (A ) k .…”

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