this book is the second revised and expanded edition of "matrix iterative analysis" by richard s. varga. the first edition was published in 1962 by prentice-hall, inc., and went out of print later. varga agreed to prepare a revised version for the springer series in computational mathematics. the revision took a long time because numerical analysis has matured into a highly diverse field, making it difficult to decide what could be added. only a few items were added, requiring little additional background. these include ovals of cassini, a semi-iterative analysis of sor methods, h-matrices and weak regular splittings, ultrametric matrices, and matrix rational approximations to exp(-z). new references and exercises have been added, misprints corrected, and numerous minor improvements made. the author thanks colleagues and friends for their encouragement, and the mathematics office at springer heidelberg for their support. the book is divided into nine chapters, covering matrix properties, nonnegative matrices, basic iterative methods, successive overrelaxation methods, semi-iterative methods, elliptic difference equations, alternating-direction implicit methods, matrix methods for parabolic pdes, and estimation of acceleration parameters. each chapter includes detailed discussions, examples, and theorems, with a focus on the analysis and application of iterative methods for solving linear systems. the book is intended for researchers and students in numerical analysis and computational mathematics.this book is the second revised and expanded edition of "matrix iterative analysis" by richard s. varga. the first edition was published in 1962 by prentice-hall, inc., and went out of print later. varga agreed to prepare a revised version for the springer series in computational mathematics. the revision took a long time because numerical analysis has matured into a highly diverse field, making it difficult to decide what could be added. only a few items were added, requiring little additional background. these include ovals of cassini, a semi-iterative analysis of sor methods, h-matrices and weak regular splittings, ultrametric matrices, and matrix rational approximations to exp(-z). new references and exercises have been added, misprints corrected, and numerous minor improvements made. the author thanks colleagues and friends for their encouragement, and the mathematics office at springer heidelberg for their support. the book is divided into nine chapters, covering matrix properties, nonnegative matrices, basic iterative methods, successive overrelaxation methods, semi-iterative methods, elliptic difference equations, alternating-direction implicit methods, matrix methods for parabolic pdes, and estimation of acceleration parameters. each chapter includes detailed discussions, examples, and theorems, with a focus on the analysis and application of iterative methods for solving linear systems. the book is intended for researchers and students in numerical analysis and computational mathematics.