Nonhomogeneous matrix products (Q2781742)

The book is a research exposition the aim of which is to put together much of the basic work on nonhomogeneous matrix products, i.e., \(A_k\dots A_1\) with (at least two) different factors. The book consists of 13 chapters and an appendix in which a few classical results used in the book are given. NEWLINENEWLINENEWLINEChapter 1 is an introduction containing some preliminary definitions and remarks. In Chapter 2 the projective metric in \(\mathbb{R}^n\), Hausdorff metric and other functionals that can be used to study convergence of infinite products of matrices are explained. Chapter 3 is devoted to algebraic properties and convergence in semigroups of matrices. NEWLINENEWLINENEWLINEChapter 4 deals with some nonnegative matrices (i.e., having nonnegative entries) and semigroups of such matrices. Ergodicity is studied in Chapter 5. Here ergodicity concerns sequences of products \(A_1\), \(A_2A_1\), \dots, which appear more like rank 1 matrices as \(k\rightarrow \infty\). NEWLINENEWLINENEWLINEIn Chapter 6 some basic convergence results for infinite products of matrices are given, in Chapter 7 continuous convergence and some sequence spaces are studied. A matrix \(A\) is called paracontracting (PC) if \(\|Ax\|<\|x\|\) whenever \(Ax\neq x\). In Chapter 8 convergence of infinite products of PC matrices is studied. In Chapters 9 and 10 a matrix set \(\Sigma\) is considered and the convergence in Hausdorff metric of \(\Sigma,\Sigma^2\dots\) is studied. In Chapter 12 slowly varying matrix products are considered. NEWLINENEWLINENEWLINEIn Chapters 11 and 13 some applications are shown. These include, e.g., the construction of curves and fractals, solving demographic problems, applications in production systems and management structures. MATLAB codes for solving some of those problems are given in the ends of Chapters 11 and 13.