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The paper suggests and illustrates a simple unified approach to deriving upper bounds for the dominant eigenvalues of the so-called PF matrices (or matrices with the Perron–Frobenius property) from those for the Perron root of a nonnegative matrix. [ABSTRACT FROM AUTHOR]

In the paper, some inequalities interrelating the singular values of a square matrix A with complex entries with the Perron roots of the associated nonnegative matrices $ {P_A}=\left| {A\left| \circ \right|\left. A \right|} \right.\ and\ \left( {{P_A}+P_A^T} \right)/2 $ are derived. The results obtained are applied to (0, 1)-matrices. [ABSTRACT FROM AUTHOR]

The paper suggests uppe bound on the second largest eigenvalue and the sum of two largest eigenvalues of symmetric nonnegative matrices and graphs. Conditions necessary and sufficient for some of the counds to be attained are established. Special attention is paid to the subclass of matrices with zero diagonal entries and with off-diagonal entries not exceeding unity, which obviously contains the adjacency matrices of undirected graphs. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper suggests new two-sided bounds for the Perron root of a weakly irreducible nonnegative matrix, which depend on the circuits of length no less than two in the associated directed graph and, in some cases, improve Al'pin's bounds, suggested in 1995. Two approaches to deriving circuit bounds are considered. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

The paper presents new two-sided bounds for the Perron root of a block-partitioned nonnegative matrix, improving Chistyakov’s bounds. The equality cases are analyzed. As an application, new conditions sufficient for a complex matrix to be a nonsingular H-matrix are obtained. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper presents new upper and lower bounds for the singular values of rectangularmatrices explicitly involving the matrix sparsity pattern. These bounds are based on an upper bound for the Perron root of a nonnegative matrix and on the sparsity-dependent version of the Ostrowski-Brauer theorem on eigenvalue inclusion regions. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

The paper presents new upper and lower bounds for the Perron root of a nonnegative matrix in terms of the simple circuits of length not exceeding k and the simple paths of length k, 1 ≤ k ≤ n, in the directed graph of the matrix. For each k, 1 ≤ k ≤ n, these bounds are intermediate between the circuit bounds and the path-dependent bounds suggested previously, and for k = 1 and k = n they reduce to the corresponding path-dependent bounds and the circuit bounds, respectively. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

The paper suggests two-sided, upper, and lower circuit bounds for the Perron root of a nonnegative matrix, most of which are derived based on an extension of the monotonicity property of the Perron root established by Fiedler and Pták. Bibliography:9titles. [ABSTRACT FROM AUTHOR]

In this paper, bounds and inequalities for the Perron root of a nonnegative matrix, extending and complementing the classical theorems of Frobenius and Ostrowski in terms of (deleted) row and column sums, are presented. All the results are derived by using the same approach, based on the monotonicity property of the Perron root. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

Let $A^\{(l)\}$$(l=1,\backslash ldots,k)$ be $n\backslash times\; n$ nonnegative matrices with right and left Perron vectors $u^\{(l)\}$ and $v^\{(l)\}$, respectively, and let $D^\{(l)\}$ and $E^\{(l)\}$$(l=1,\backslash ldots,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $u^\{(1)\}\backslash circ\; v^\{(1)\}=\backslash ldots=u^\{(k)\}\backslash circ\; v^\{(k)\}\backslash ne0$ (where ``$\backslash circ$'' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices $A^\{(l)\}$ be irreducible, for the Perron root of the sum $\backslash sum^k\_\{l=1\}D^\{(l)\}A^\{(l)\}E^\{(l)\}$ we derive a lower bound of the form $\backslash rho\backslash left(\backslash sum^k\_\{l=1\}D^\{(l)\}A^\{(l)\}E^\{(l)\}\backslash right)\backslash ge\; \backslash sum^\{k\}\_\{l=1\}\backslash be\_l\backslash rho(A^\{(l)\}),\backslash quad\; \backslash beta\_l>0.$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^\{(l)\}$$(l=1,\backslash ldots,k)$, $\backslash rho\backslash left(\backslash sum^\{k\}\_\{l=1\}A^\{(l)\}\backslash right)\backslash ge\; \backslash sum^k\_\{l=1\}\backslash alpha\_l\backslash rho(A^\{(l)\}),$ where the coefficients $\backslash alpha\_l>0$ are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

A correction to the article "Improving Chistyakov's Bounds for the Perron Root of a Nonnegative Matrix," by L. Yu. Kolotilina that was published in the April 2008 issue is presented.