Nonsingular H-matrices have proven to be a source of many interesting results in different research areas in numerical linear algebra and also in applications in economy, engineering, ecology. In this paper, a new criteria for identifying some special H-matrices is presented. It is based on attributing different partition of the index set to each row of the matrix in consideration and testing inequalities that are associated to these partitions and involve recursively defined Nekrasov row sums. The new subclass of H-matrices introduced in this way is then analyzed with respect to its relation to well-known matrix classes. We used our new condition to estimate norm of the inverse matrix and errors in linear complementarity problems that involve matrices of this type.

In this paper, we consider pi-Nekrasov matrices, a generalization of {P1, P2}-Nekrasov matrices obtained by introducing the set pi = {P1, P2, ..., Pm} of m simultaneous permutations of rows and columns of the given matrix. For point-wise and block pi-Nekrasov matrices we give infinity norm bounds for the inverse. For pi-Nekrasov B-matrices, obtained through a special rank one perturbation, we present main results on infinity norm bounds for the inverse and error bounds for linear complementarity problems. Numerical examples illustrate the benefits of new bounds.

In this paper, we consider ??Nekrasov matrices, a generalization of {P1, P2}?Nekrasov matrices obtained by introducing the set ? = {P1, P2, ..., Pm} of m simultaneous permutations of rows and columns of the given matrix. For point-wise and block ??Nekrasov matrices we give infinity norm bounds for the inverse. For ??Nekrasov B?matrices, obtained through a special rank one perturbation, we present main results on infinity norm bounds for the inverse and error bounds for linear complementarity problems. Numerical examples illustrate the benefits of new bounds.

{P1,P2}-Nekrasov matrices represent a generalization of Nekrasov matrices via permutations. In this paper, we obtained an error bound for linear complementarity problems for fP1; P2g-Nekrasov matrices. Numerical examples are given to illustrate that new error bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of {P1,P2}-Nekrasov matrices in the block case, considering two different types of block generalizations. Numerical examples show that new norm bounds for the block case can give tighter results compared to already known bounds for the point-wise case.

Lower-semi-Nekrasov matrices represent a generalization of Nekrasov matrices. For the inverse of lower-semi-Nekrasov matrices, a max-norm bound is proposed. Numerical examples are given to illustrate that new norm bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of lower-semi-Nekrasov matrices in the block case. We considered two types of block generalizations and illustrated the results with numerical examples.

In this paper, we consider the class of PH−matrices, a subclass of H−matrices and, using scaling characterization, we show that this class is closed under taking the Schur complement. We show that, under certain conditions, the Perron complement of a PH−matrix is a PH−matrix. We also present a way of constructing a scaling matrix for the given PH−matrix and we give eigenvalue localization for the Schur complement of a PH−matrix using only the entries of the original matrix. We illustrate this by numerical examples.

The theory of M- and H-matrices has become one of the basic tools in applied linear algebra and it has contributed to different areas of mathematical research and applications. Many results in numerical analysis, eigenvalue localization problems, analysis of iterative methods for solving systems of linear equations came from H-matrix theory. Also, many results in engineering rely on mathematical foundation that is, explicitly or implicitly, formulated in terms of H-matrices. In this talk, different matrix properties that guarantee nonsingularity of matrices and define different subclasses of H-matrices will be presented together with related results concerning Schur complement matrices, eigenvalue localization and bounds of the max-norm of the inverse matrix.

It is well-known that for a given H-matrix A there exists a diagonal nonsingular matrix that scales A (by multiplying it from the right) to a strictly diagonally dominant (SDD) matrix. There are subclasses of H-matrices that can be fully characterised by the form of the corresponding diagonal scaling matrices. However, for some applications, it is not necessary to have such full characterisation. It is sufficient to find at least one scaling matrix that will do the job. The aim of this paper is to present a way of constructing a diagonal scaling matrix for one special subclass of H-matrices called Partition-Nekrasov matrices. As an application of this scaling approach, we obtain eigenvalue localisation for the corresponding Schur complement matrix, using only the entries of the original matrix.

In this paper we present a nonsingularity result which is a generalization of Nekrasov property by using two different permutations of the index set. The main motivation comes from the following observation: matrices that are Nekrasov matrices up to the same permutations of rows and columns, are nonsingular. But, testing all the permutations of the index set for the given matrix is too expensive. So, in some cases, our new nonsingularity criterion allows us to use the results already calculated in order to conclude that the given matrix is nonsingular. Also, we present new max-norm bounds for the inverse matrix and illustrate these results by numerical examples, comparing the results to some already known bounds for Nekrasov matrices.

The theory of Schur complement plays an important role in many fields, as well as the theory of H-matrices. In this paper, we obtain some bounds for the eigenvalues of Schur complement by the entries of the original matrix instead of those of the Schur complement, for some special H-matrices. Also, we show how this result for a wider class can be applied for estimating bounds for the eigenvalues of Schur complement of an SDD matrix.

It is well known, see [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246–251 [2]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009–1030] [14], that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant, as well as its diagonal-Schur complement. Also, if a matrix is an H-matrix, then its Schur complement and diagonal-Schur complement are H-matrices, too, see [J. Liu, Y. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Alg. Appl. 389 (2004) 365–380] [13]. Recent research, see [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Alg. Appl. 378 (2004) 231–244 [12]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009–1030] [14], showed that the similar statements hold for some special subclasses of H-matrices. The aim of this paper is to give more invariance results of this type, and simplified proofs for some already known results, by using scaling approach.