Separable Type Representations of Matrices and Fast Algorithms : Volume 1 Basics. Completion Problems. Multiplication and Inversion Algorithms Part 1. Basics on separable semiseparable and quasiseparable representations of matrices 1. Matrices with separable representation and low complexity algorithms 2. The minimal rank completion problem 3. Matrices in diagonal plus semiseparable form 4. Quasiseparable representations: the basics 5. Quasiseparable generators 6. Rank numbers of pairs of mutually inverse matrices Asplund theorems 7. Unitary matrices with quasiseparable representations Part 2. Completion of matrices with specified bands 8. Completion to Green matrices 9. Completion to matrices with band inverses and with minimal ranks 10. Completion of special types of matrices 11. Completion of mutually inverse matrices 12. Completion to unitary matrices Part 3. Quasiseparable representations of matrices descriptor systems with boundary conditions and first applications 13. Quasiseparable representations and descriptor systems with boundary conditions 14. The first inversion algorithms 15. Inversion of matrices in diagonal plus semiseparable form 16. Quasiseparable semiseparable representations and one direction systems 17. Multiplication of matrices Part 4. Factorization and inversion 18. The LDU factorization and inversion 19. Scalar matrices with quasiseparable order one 20. The QR factorization based method This two volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The primary focus is on fast algorithms (many of linear complexity) for matrices in semiseparable quasiseparable band and companion form. The work examines algorithms of multiplication inversion and description of eigenstructure and includes a wealth of illustrative examples throughout the different chapters. The first volume consists of four parts. The first part is mainly theoretical in character introducing and studying the quasiseparable and semiseparable representations of matrices and minimal rank completion problems. Three further completions are treated in the second part. The first applications of the quasiseparable and semiseparable structure are included in the third part where the interplay between the quasiseparable structure and discrete time varying linear systems with boundary conditions play an essential role. The fourth part includes factorization and inversion fast algorithms for matrices via quasiseparable and semiseparable structures. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and accessible style the text will be a valuable resource for engineers scientists numerical analysts computer scientists and mathematicians alike