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A simple proof for the inequality between the perron root of a nonnegative matrix and that of its geometric symmetrization
AbstractLet A = (aij) be a nonnegative square matrix, let G = (gij) be its geometric symmetrization, i.e., gij = $$ \sqrt {a_{ij} a_{ji} } $$, and let ρ denote the Perron root. We present a simpleExpand
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Combinatorial properties of irreducible semigroups of nonnegative matrices
The paper suggests a combinatorial proof of the Protasov–Voynov theorem on an irreducible semigroup of nanonegative matrices free of positive matrices. This solves the problem posed by the authors ofExpand
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Combinatorial Properties of Entire Semigroups of Nonnegative Matrices
Generalizations of the Protasov–Voynov theorem on the structure of irreducible semigroups of nonnegative matrices free of zero rows and columns are obtained. The theorem is extended to semigroupsExpand
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The generalized monotonicity property of the Perron root
The paper presents a new monotonicity property of the Perron root of a nonnegative matrix. It is shown that this new property implies known monotonicity properties and also the Chistyakov two-sidedExpand
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On the Simultaneous Triangulability of Matrices
Two necessary and sufficient criteria for the simultaneous triangulability of two complex matrices are established. Both of them admit a finite verification procedure. To prove the first criterion,Expand
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A New Proof of the Protasov-Voynov Theorem on Semigroups of Nonnegative Matrices
A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods.
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Combinatorial structure of k-semiprimitive matrix families
Protasov's Theorem on the combinatorial structure of k-primitive families of non-negative matrices is generalized to k-semiprimitive matrix families. The main tool is the binary relation of colourExpand
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Bounds for joint spectral radii of a set of nonnegative matrices
Bounds for joint spectral radii of a set of nonnegative matrices are established by using the apparatus of idempotent algebra.
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