Introduction

The definition of a complex-valued matrix derivative was given in Chapter 3 (see Definition 3.1). In this chapter, it will be shown how the complex-valued matrix derivatives can be found for all nine different types of functions given in Table 2.2. Three different choices are given for the complex-valued input variables of the functions, namely, scalar, vector, or matrix; in addition, three possibilities for the type of output that functions return, again, could be scalar, vector, or matrix. The derivative can be identified through the complex differential by using Table 3.2. In this chapter, it will be shown how the theory introduced in Chapters 2 and 3 can be used to find complex-valued matrix derivatives through examples. Many results are collected inside tables to make them more accessible.

The rest of this chapter is organized as follows: The simplest case, when the output of a function is a complex-valued scalar, is treated in Section 4.2, which contains three subsections (4.2.1, 4.2.2, and 4.2.3) when the input variables are scalars, vectors, and matrices, respectively. Section 4.3 looks at the case of vector functions; it contains Subsections 4.3.1, 4.3.2, and 4.3.3, which treat the three cases of complex-valued scalar, vector, and matrix input variables, respectively. Matrix functions are considered in Section 4.4, which contains three subsections. The three cases of complex-valued matrix functions with scalar, vector, and matrix inputs are treated in Subsections 4.4.1, 4.4.2, and 4.4.3, respectively. The chapter ends with Section 4.5, which consists of 10 exercises.