Hessians of scalar functions of complex-valued matrices: A systematic computational approach

A systematic theory is introduced for finding the four Hessians of complex-valued scalar functions with respect to a complex-valued matrix variable and the complex conjugate of this variable. It is shown how the four Hessian matrices of a scalar complex function can be identified from the second-order complex differential of the scalar function. These Hessians are the four parts of a bigger matrix which must be checked in order to identify if a stationary point is a local minimum, maximum, or saddle point. The method introduced is general such that many results can be derived using the framework introduced. Hessians are derived for some useful examples taken from signal processing related functions.