New scaling criteria for $H$-matrices and applications

Non-singular $H$-matrices represent an important research frame in analysis of many classical problems of numerical linear algebra, as well as in applications in engineering, health, information sciences, and social studies. As identification of $H$-matrices was never an easy task, a research area was formed around some special $H$-matrices, characterized by checkable conditions-inequalities expressed via matrix entries only. In this paper, we introduced new conditions for a given matrix to be a non-singular $H$-matrix. We introduced a new special subclass of non-singular $H$-matrices and applied new criterion to obtain results on infinity norm of the inverse matrix, errors in linear complementarity problems, and estimation of minimal singular value. Also, results on spectra of the Schur complement matrix were given in the form of scaled disks and in the form of intervals that included or excluded real parts of eigenvalues. Results were interpreted in the light of mixed linear complementarity problems. Numerical examples illustrated improvements obtained by applications of new criteria.