The Laurent expansion for a nearly singular matrix

Langenhop, C. E.

doi:10.1016/0024-3795(71)90004-8

Cited by 47 publications

(31 citation statements)

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“…Using results there, we show that v ind A in (1.1) and obtain extensions of some of the results in [ [3] in the form to which they reduce when A o A and A1 I. Specific references in [3] are given in parentheses c.E.…”

supporting

confidence: 51%

“…Specific references in [3] are given in parentheses c.E. LANGENHOP although some of the results quoted here can be proved relatively easily without the more complicated machinery used for the general case.…”

mentioning

confidence: 97%

“…where Q e C "" for k >__ v, Q_ 4: 0, and the series converges (elementwise) for 0 < Izl < p, In [3] we considered for A o, A e C" " the existence of the Laurent expansion of (Ao + zA)-1 in powers of z in a deleted neighborhood of z 0.…”

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confidence: 99%

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On the Index of a Square Matrix

Langenhop¹

1971

SIAM J. Appl. Math.

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Let A be a nonzero matrix of complex numbers and let be the corresponding identity. The index of A (the least nonnegative integer k such that rank A rank A 1) is related to the least integer m such that limz-.O z I(A + zl)-1A exists (z being a complex variable). It is shown that the two are equal if the index of A is at least two; the latter is one if the index of A is zero or one. It is also shown that if b is a nonzero n x matrix of complex numbers and the index of A equals v, then b is in the range of A if and only if limz-o (A + zI)-lb exists.

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confidence: 51%

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On the Index of a Square Matrix

Langenhop¹

1971

SIAM J. Appl. Math.

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“….) are uniquely defined by the following properties : Theorem 1: [1] With (zE − A) regular and Φ i defined by (2), the following properties between the fundamental matrices Φ i holds :…”

Section: Preliminary Resultsmentioning

confidence: 96%

“…Definition 16: The regular system (1) is controllable to the origin in the forward sense if, for every x 0 ∈ H Iu , there exists a finite k > 0 and a control u 0,k+μ such that x (k) = 0.…”

Section: Rank (A + B) = Rank a Bmentioning

confidence: 99%

Reachability and controllability of discrete time descriptor systems

Gregoriadou

Karampetakis

2012

Ccca12

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The causal and non-causal reachability Gramians are used, in order to define the reachability subspace which is the set of all states reachable from the origin. Controllability of the disrete-time regular descriptor system by means of the forward solution is examined and new criteria for testing reachability (controllability from the origin) and controllability to the origin in the forward sense are proved.

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On the Hamiltonicity Gap and doubly stochastic matrices

Borkar

Ejov

Filar

2008

Random Struct Algorithms

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We consider the Hamiltonian cycle problem embedded in singularly perturbed (controlled) Markov chains. We also consider a functional on the space of stationary policies of the process that consists of the (1,1)-entry of the fundamental matrices of the Markov chains induced by these policies. We focus on the subset of these policies that induce doubly stochastic probability transition matrices which we refer to as the "doubly stochastic policies". We show that when the perturbation parameter, ε, is sufficiently small, the minimum of this functional over the space of the doubly stochastic policies is attained at a Hamiltonian cycle, provided that the graph is Hamiltonian. We also show that when the graph is non-Hamiltonian, the above minimum is strictly greater than that in a Hamiltonian case. We call the size of this difference the "Hamiltonicity Gap" and derive a conservative lower bound for this gap. Our results imply that the Hamiltonian cycle problem is equivalent to the problem of minimizing the variance of the first hitting time of the home node, over doubly stochastic policies.

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The Laurent expansion for a nearly singular matrix

Cited by 47 publications

References 1 publication

On the Index of a Square Matrix

On the Index of a Square Matrix

Reachability and controllability of discrete time descriptor systems

On the Hamiltonicity Gap and doubly stochastic matrices

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