This book is written as an engineering-oriented mathematics book. It introduces the field involved in finding derivatives of complex-valued functions with respect to complex-valued matrices, in which the output of the function may be a scalar, a vector, or a matrix. The theory of complex-valued matrix derivatives, collected in this book, will benefit researchers and engineers working in fields such as signal processing and communications. Theories for finding complex-valued derivatives with respect to both complex-valued matrices with independent components and matrices that have certain dependencies among the components are developed and illustrative examples that show how to find such derivatives are presented. Key results are summarized in tables. Through several research-related examples, it will be shown how complex-valued matrix derivatives can be used as a tool to solve research problems in the fields of signal processing and communications.

This book is suitable for M.S. and Ph.D. students, researchers, engineers, and professors working in signal processing, communications, and other fields in which the unknown variables of a problem can be expressed as complex-valued matrices. The goal of the book is to present the tools of complex-valued matrix derivatives such that the reader is able to use these theories to solve open research problems in his or her own field. Depending on the nature of the problem, the components inside the unknown matrix might be independent, or certain interrelations might exist among the components. Matrices with independent components are called unpatterned and, if functional dependencies exist among the elements, the matrix is called patterned or structured.