The Complex Gradient Operator and the CR-Calculus

TL;DR

This paper develops a comprehensive calculus framework for real-valued functions of complex vectors using Wirtinger Calculus, including second-order derivatives, aimed at electrical engineering education.

Contribution

It extends the existing first-order complex vector calculus to include detailed second-order considerations, filling a pedagogic gap.

Findings

Provides a thorough development of complex gradient and Hessian calculus

Suitable for teaching complex derivatives in electrical engineering

Includes detailed second-order derivative analysis

Abstract

A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus. The presented material is suitable for exposition in an introductory Electrical Engineering graduate level course on the use of complex gradients and complex Hessian matrices, and has been successfully used in teaching at UC San Diego. Going beyond the commonly encountered treatments of the first-order complex vector calculus, second-order considerations are examined in some detail filling a gap in the pedagogic literature.