Introduction to the Book

To solve increasingly complicated open research problems, it is crucial to develop useful mathematical tools. Often, the task of a researcher or an engineer is to find the optimal values of unknown parameters that can be represented by complex-valued matrices. One powerful tool for finding the optimal values of complex-valued matrices is to calculate the derivatives with respect to these matrices. In this book, the main focus is on complex-valued matrix calculus because the theory of real-valued matrix derivatives has been thoroughly covered already in an excellent manner in Magnus and Neudecker (1988). The purpose of this book is to provide an introduction to the area of complex-valued matrix derivatives and to show how they can be applied as a tool for solving problems in signal processing and communications.

The framework of complex-valued matrix derivatives can be used in the optimization of systems that depend on complex design parameters in areas where the unknown parameters are complex-valued matrices with independent components, or where they belong to sets of matrices with certain structures. Many of the results discussed in this book are summarized in tabular form, so that they are easily accessible. Several examples taken from recently published material show how signal processing and communication systems can be optimized using complex-valued matrix derivatives. Note that the differentiation procedure is usually not sufficient to solve such problems completely; however, it is often an essential step toward finding the solution to the problem.

In many engineering problems, the unknown parameters are complex-valued matrices, and often, the task of the system designer is to find the values of these complex parameters, which optimize a certain scalar real-valued objective function.