Patterned complex-valued matrix derivatives

A systematic and simple method is proposed for how to find the derivative of complex-valued matrix functions which depend on matrix arguments that contain patterns. The proposed method is developed by means of the chain rule and it is able to handle both linear and nonlinear patterns. One main issue of the proposed method is to identify a matrix function which depends on independent variables and its domain must have the same dimension as the dimension of the set of patterned matrices. In addition, this function must produce all the matrices within the pattern of interest. Some illustrative examples which are relevant for problems in the area of signal processing for communications are presented.