Stability of matrices with negative diagonal submatrices

Nieuwenhuis, Herman J.; Schoonbeek, Lambert

doi:10.1016/s0024-3795(02)00304-x

Cited by 9 publications

(24 citation statements)

References 2 publications

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“…In addition, based on Sylvester's theorem [25][26][27], the number of negative diagonal elements of matrix ( 5) is equal to the number of negative eigenvalues of this matrix. Since all diagonal elements (5), taking into account ( 2) and (3), are negative, all roots of the characteristic equation have negative real parts.…”

Section: Analysis Of the Characteristic Polynomial Rootsmentioning

confidence: 99%

Transient Behavior of the MAP/M/1/N Queuing System

2021

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This paper investigates the characteristics of the MAP/M/1/N queuing system in the transient mode. The matrix method for solving the Kolmogorov equations is proposed. This method makes it possible, in general, to obtain the main characteristics of the considered queuing system in a non-stationary mode: the probability of losses, the time of the transient mode, the throughput, and the number of customers in the system at time t. The developed method is illustrated by numerical calculations of the characteristics of the MAP/M/1/3 system in the transient mode.

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Section: Analysis Of the Characteristic Polynomial Rootsmentioning

confidence: 99%

Transient Behavior of the MAP/M/1/N Queuing System

2021

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“…It is well-known that System (2) is asymptotically stable if and only if the matrix C is stable, i.e. all its eigenvalues have negative real parts (see [197]). The following sufficient for stability conditions were established in [197] (see [197], Theorems 2 and 3, Corollary 1.…”

Section: Example Of Application To Dynamical System Stabilitymentioning

confidence: 99%

“…all its eigenvalues have negative real parts (see [197]). The following sufficient for stability conditions were established in [197] (see [197], Theorems 2 and 3, Corollary 1. For the definition of negative diagonally dominant (NDD) matrices, see Appendix).…”

Section: Example Of Application To Dynamical System Stabilitymentioning

confidence: 99%

“…Criterion 1 ( [197]) Let A = {a ij } n i,j=1 and B = {b ij } n i,j=1 be real n × n matrices, and the 2n × 2n matrix C be defined by (3). Let a ii < 0 and b ii < 0 for all i = 1, .…”

Section: Example Of Application To Dynamical System Stabilitymentioning

confidence: 99%

“…B be negative diagonal) and A is NDD, then C is stable. Criterion 2 ( [197]) Let A = {a ij } n i,j=1 be a real n × n matrix with a ii < 0 for all i = 1, . .…”

Section: Example Of Application To Dynamical System Stabilitymentioning

confidence: 99%

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Unifying Matrix Stability Concepts with a View to Applications

Kushel¹

2019

SIAM Rev.

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Multiplicative and additive D-stability, diagonal stability, Schur Dstability, H-stability are classical concepts which arise in studying linear dynamical systems. We unify these types of stability, as well as many others, in one concept of (D, G, •)-stability, which depends on a stability region D ⊂ C, a matrix class G and a binary matrix operation •. This approach allows us to unite several well-known matrix problems and to consider common methods of their analysis. In order to collect these methods, we make a historical review, concentrating on diagonal and D-stability. We prove some elementary properties of (D, G, •)-stable matrices, uniting the facts that are common for many partial cases. Basing on the properties of a stability region D which may be chosen to be a concrete subset of C (e.g. the unit disk) or to belong to a specified type of regions (e.g. LMI regions) we briefly describe the methods of further development of the theory of (D, G, •)-stability. We mention some applications of the theory of (D, G, •)-stability to the dynamical systems of different types.

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Novel Versions of D-Stability in Matrices Provide New Insights into ODE Dynamics

Kushel

Pavani

2023

Mediterr. J. Math.

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Stability of matrices with negative diagonal submatrices

Cited by 9 publications

References 2 publications

Transient Behavior of the MAP/M/1/N Queuing System

Transient Behavior of the MAP/M/1/N Queuing System

Unifying Matrix Stability Concepts with a View to Applications

Novel Versions of D-Stability in Matrices Provide New Insights into ODE Dynamics

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