Finding patterned complex-valued matrix derivatives by using manifolds

Often in engineering, the design requirements are to find a complex-valued matrix which minimizes or maximizes a real-valued objective function under the constraint that the matrix belongs to a set of matrices with pattern. Recently, a systematic method was published for finding the derivative of complex-valued matrix functions which depend on matrix arguments that contain patterns. Central in this theory is the pattern producing function. Derivatives with respect to the input parameters of the pattern producing function were proposed earlier. Now, slightly stricter requirements are put on the pattern producing function such that explicit expressions can be found for the patterned derivatives with respect the actual patterned matrices. Several examples are presented.