Generalized band matrices and their inverses

The idea of defining the generalized band matrices is based on the recognition that several pattern matrices and their inverses have low rank submatrices in certain domains. Theoretical considerations concerning the generalized band matrices enable us to give uniform treatment for several well known classes of matrices like band matrices, block band matrices, band matrices with low rank corrections, sparse matrices and their inverses. Making use of the new notions of information content and of compact representation of matrices, the concept of proper matrices is extended for generalized band matrices. Some reduction algorithms are presented which help to discover certain hidden structural properties of the generalized band matrices. The theoretical results are enlightened by a large number of figures illustrating numerical examples. Work supported by the Progetto Finalizzato Calcolo Parallelo e Sistemi Informatici of CNR. Visiting Professor at the University of Pisa under the support of GNIM-CNR.