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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]

The new upper boundλmaxA≤∑k=1p+1i≡kmodp+1maxλmaxAiifor the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = (Aij ) of block semibandwidth p is suggested. In the special case where the diagonal blocks of A are identity matrices, the latter bound reduces to the boundλmaxA≤p+1,depending on p only, which improves the bounds established for such matrices earlier and extends the boundλmaxA≤2,old known for p = 1, i.e., for block tridiagonal matrices, to the general case p ≥ 1. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases. [ABSTRACT FROM AUTHOR]

The paper suggests upper bounds for the l norm of the inverses to block matrices belonging to certain subclasses of the class of block $$ \mathrm{\mathscr{H}} $$-matrices, which improve and supplement known results. [ABSTRACT FROM AUTHOR]

Two-sided bounds on | det A| for Nekrasov and S-Nekrasov matrices A are obtained. It is shown that for Nekrasov matrices the new bounds improve the known bounds of Bailey and Crabtree. As to the relatively recently introduced S-Nekrasov matrices, no bounds on their determinants have been suggested, as far as the author is aware. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles. [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]