The only book devoted exclusively to matrix functions, this research monograph gives a thorough treatment of the theory of matrix functions and numerical methods for computing them. The author's elegant presentation focuses on the equivalent definitions of f(A) via the Jordan canonical form, polynomial interpolation, and the Cauchy integral formula, and features an emphasis on results of practical interest and an extensive collection of problems and solutions. Functions of Matrices more than just a monograph on matrix functions; its wide-ranging content--including an overview of applications, historical references, and miscellaneous results, tricks, and techniques with an f(A) connection--makes it useful as a general reference in numerical linear algebra. Other key features of the book include development of the theory of conditioning and properties of the Frchet derivative; an emphasis on the Schur decomposition, the block Parlett recurrence, and judicious use of Pad approximants; the inclusion of new, unpublished research results and improved algorithms; a chapter devoted to the f(A)b problem; and a MATLAB toolbox providing implementations of the key algorithms.