This book, "Non-negative Matrices and Markov Chains," is the second edition of a previously published work by E. Seneta. It is part of the Springer Series in Statistics, with advisors including D. Brillinger, S. Fienberg, J. Gani, J. Hartigan, J. Kiefer, K. Krickeberg, and E. Seneta. The book provides a comprehensive treatment of non-negative matrices and their connection to Markov chains, covering both finite and infinite matrices. It is aimed at mathematicians and professionals in applied fields such as probability theory, numerical analysis, demography, mathematical economics, and dynamic programming. The mathematical prerequisites include knowledge of real-variable theory, matrix theory, and some complex-variable theory, with an emphasis on real-variable methods. The first four chapters focus on finite non-negative matrices, while the following three chapters develop an analogous theory for infinite matrices. The book is structured to allow readers to focus on specific areas of interest, with chapters containing varying degrees of interdependence. Chapter 1 provides foundational concepts, while subsequent chapters build upon this foundation. Chapters 2–4 are more interdependent, and Chapter 5 is recommended before Chapters 6 and 7 for readers interested in infinite matrices. The book includes exercises that are closely tied to the text, offering further development of the theory and deeper insights. It has been modified from the author's earlier work, "Non-negative Matrices," with new sections and expanded content, particularly in the areas of inhomogeneous products of non-negative matrices and the computation of stationary distributions of infinite Markov chains. The book also includes a glossary of notation and symbols, appendices on number theory, matrix lemmas, and upper semi-continuous functions, along with a bibliography and indexes.This book, "Non-negative Matrices and Markov Chains," is the second edition of a previously published work by E. Seneta. It is part of the Springer Series in Statistics, with advisors including D. Brillinger, S. Fienberg, J. Gani, J. Hartigan, J. Kiefer, K. Krickeberg, and E. Seneta. The book provides a comprehensive treatment of non-negative matrices and their connection to Markov chains, covering both finite and infinite matrices. It is aimed at mathematicians and professionals in applied fields such as probability theory, numerical analysis, demography, mathematical economics, and dynamic programming. The mathematical prerequisites include knowledge of real-variable theory, matrix theory, and some complex-variable theory, with an emphasis on real-variable methods. The first four chapters focus on finite non-negative matrices, while the following three chapters develop an analogous theory for infinite matrices. The book is structured to allow readers to focus on specific areas of interest, with chapters containing varying degrees of interdependence. Chapter 1 provides foundational concepts, while subsequent chapters build upon this foundation. Chapters 2–4 are more interdependent, and Chapter 5 is recommended before Chapters 6 and 7 for readers interested in infinite matrices. The book includes exercises that are closely tied to the text, offering further development of the theory and deeper insights. It has been modified from the author's earlier work, "Non-negative Matrices," with new sections and expanded content, particularly in the areas of inhomogeneous products of non-negative matrices and the computation of stationary distributions of infinite Markov chains. The book also includes a glossary of notation and symbols, appendices on number theory, matrix lemmas, and upper semi-continuous functions, along with a bibliography and indexes.