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The class of quasiseparable matrices includes several other well-known matrices, such as band matrices, diagonal plus semiseparable matrices, tridiagonal matrices, and unitary Hessenberg matrices; and arises naturally in classical control theory. As with any matrix operator, computational benefits can be obtained from factoring, especially by utilizing unitary factors. This book provides some detail concerning a current (largely unitary) approach to factoring quasiseparable matrices described by Y. Eidelman and I. Gohberg (2002), derived in part from a previous description by P. M. Dewilde and A. J. van der Veen (1998). In addition to the discussion of the factorization process, this book concludes with an investigation and discussion of the stability of the factorization process.