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This book provides a first detailed and self-contained account of linear-algebraic aspects of max-algebra for general matrices. Among the main features of the proposed book is the presentation of the fundamental max-algebraic theory, in one place in a comprehensive and unified form, made with all proofs and in full generality (that is for both irreducible and reducible matrices). Advanced material within the book also provides the reader with a wealth of information that has never been published before.