Journal ArticleDOI
On the singular graph and the Weyr characteristic of anM-matrix
TLDR
In this paper, the authors define the singular graph of an M-matrix in standard lower block triangular form, with diagonal blocks irreducible, as the set of indices α such that the diagonal block Aαα is singular.Abstract:
LetA be anM-matrix in standard lower block triangular form, with diagonal blocksAii irreducible. LetS be the set of indices α such that the diagonal blockAαα is singular. We define the singular graph ofA to be the setS with partial order defined by α > β if there exists a chain of non-zero blocksAαi, Aij, ⋯, Alβ.read more
Citations
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The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey
TL;DR: For a nonnegative matrix P, the relation of its marked reduced graph to that part of the Jordan form associated with the Perron-Frobenius root is discussed in this paper.
Journal ArticleDOI
Theorems on M-splittings of a singular M-matrix which depend on graph structure☆
TL;DR: In this paper, the spectral properties of the iteration matrix M-1N were investigated by considering the relationships of the graphs of A, M, N, and M 1 n for an irreducible Z-matrix A, and it was shown that the circuit index of M -1N is the greatest common divisor of certain sets of integers associated with the circuits of A.
Journal ArticleDOI
Ranks of zero patterns and sign patterns
TL;DR: In this article, a sufficient condition on P such that all matrices over F with pattern P have the same height characteristic was shown to be necessary and sufficient for the nonsingularity of a sign pattern.
Journal ArticleDOI
The Growth of Powers of a Nonnegative Matrix
Shmuel Friedland,Hans Schneider +1 more
TL;DR: This paper analyses the asymptotic behavior of each entry of A to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.
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The concepts of irreducibility and full indecomposability of a matrix in the works of Frobenius, König and Markov
TL;DR: In this article, a theorem which characterizes irreducible and fully indecomposable matrices in an algebraic manner was proved, based on the Frobenius-Konig theorem.
References
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Book
The Theory of Matrices
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Book
Matrix iterative analysis
TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.