Introduction

Often in signal processing and communications, problems appear for which we have to find a complex-valued matrix that minimizes or maximizes a real-valued objective function under the constraint that the matrix belongs to a set of matrices with a structure or pattern (i.e., where there exist some functional dependencies among the matrix elements). The theory presented in previous chapters is not suited for the case of functional dependencies among elements of the matrix. In this chapter, a systematic method is presented for finding the generalized derivative of complex-valued matrix functions, which depend on matrix arguments that have a certain structure. In Chapters 2 through 5, theory has been presented for how to find derivatives and Hessians of complex-valued functions F: ℂN×Q × ℂN×Q → ℂM×P with respect to the complex-valued matrix Z∈ℂN×Q and its complex conjugate Z* ℂN×Q. As seen from Lemma 3.1, the differential variables d vec(Z) and d vec(Z*) should be treated as independent when finding derivatives. This is the main reason why the function F: ℂN×Q × ℂN×Q → ℂM×Pis denoted by two complex-valued input arguments F(Z, Z*) because Z ∈ ℂN×Q and Z* ∈ ℂN×Q should be treated independently when finding complex-valued matrix derivatives (see Lemma 3.1). Based on the presented theory, up to this point, it has been assumed that all elements of the input matrix variable Z contain independent elements.