A combinatorial converse to the Perron-Frobenius theorem

Eschenbach, Carolyn A.; Johnson, Charles R.

doi:10.1016/0024-3795(90)90026-9

Cited by 19 publications

(16 citation statements)

References 3 publications

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“…As it is remarked in [3], the sign-patterns P whose signed digraphs have a positive cycle are exactly those that allow the spectral radius be an eigenvalue. In other words, allowing a positive eigenvalue is equivalent to allowing the spectral radius be an eigenvalue.…”

mentioning

confidence: 77%

“…The classical result that the spectral radius is an eigenvalue of any (entrywise) nonnegative matrix follows readily. A combinatorial converse of the latter is shown in Eschenbach and Johnson [3], where the authors prove that the sign-pattern class of a matrix requires the spectral radius to be an eigenvalue if and only if all the cycles in its signed digraph are positive.…”

Section: Introductionmentioning

confidence: 97%

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Sign-patterns which require a positive eigenvalue

Kirkland

McDonald

Tsatsomeros

1996

Linear and Multilinear Algebra

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We investigate matrices which have a positive eigenvalue by virtue of their sign-pattern and regardless of the magnitudes of the entries. When all the off-diagonal entries are nonzero, we show that an n × n sign-pattern, n = 3, 4, requires a positive eigenvalue if and only if it has at least one nonnegative diagonal entry and every cycle of length greater than one in its signed digraph is positive. When n = 3, 4, or when not all off-diagonal entries are nonzero, positivity of the cycles of length greater than one is no longer necessary. In the course of proving these results we observe certain necessary and certain sufficient *

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mentioning

confidence: 77%

Section: Introductionmentioning

confidence: 97%

Sign-patterns which require a positive eigenvalue

Kirkland

McDonald

Tsatsomeros

1996

Linear and Multilinear Algebra

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“…In [7], as we already mentioned, the Perron property stands for having the spectral radius as an eigenvalue, whereas, in [11] the weak Perron property stands for having the spectral radius as a simple positive and strictly dominant eigenvalue. In [19], the Perron-Frobenius property stands for having the spectral radius as a positive eigenvalue with a nonnegative eigenvector; this is also the definition used in [6].…”

Section: Elamentioning

confidence: 95%

On general matrices having the Perron-Frobenius Property

Elhashash¹,

Szyld²

2008

ELA

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Abstract.A matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are proved. Similarity transformations leaving such sets invariant are completely described, and it is shown that a nonnilpotent matrix eventually capturing the Perron-Frobenius property is in fact a matrix that already has it.

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“…Details regarding the following lemma can be found in [2] and in [3]. Lemma 1.3 Let A ∈ M n (IR) be an irreducible matrix.…”

Section: Introductionmentioning

confidence: 99%

Conditions for the positivity of determinants

Johnson

Neumann

Tsatsomeros

1996

Linear and Multilinear Algebra

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Consider a matrix with positive diagonal entries, which is similar via a positive diagonal matrix to a symmetric matrix, and whose signed directed graph has the property that if a cycle and its symmetrically placed complement have the same sign, then they are both positive. We provide sufficient conditions so that A be a P-matrix, that is, a matrix whose principal minors are all positive. We further provide sufficient conditions for an arbitrary matrix A whose (undirected) graph is subordinate to a tree, to be a P-matrix. If, in addition, A is sign symmetric and its undirected graph is a tree, we obtain necessary and sufficient conditions that it be a P-matrix. We go on to consider the positive semi-definiteness of symmetric matrices whose graphs are subordinate to a given tree and discuss the convexity of the set of all such matrices.

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A combinatorial converse to the Perron-Frobenius theorem

Cited by 19 publications

References 3 publications

Sign-patterns which require a positive eigenvalue

Sign-patterns which require a positive eigenvalue

On general matrices having the Perron-Frobenius Property

Conditions for the positivity of determinants

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