The Theory of Quaternion Matrix Derivatives

TL;DR

This paper develops a comprehensive and correct framework for calculating derivatives of quaternion matrix functions, addressing previous inaccuracies and enabling direct computation useful in signal processing applications.

Contribution

It introduces a systematic, rule-based methodology for quaternion matrix derivatives that corrects prior flaws and applies to both analytic and nonanalytic functions.

Findings

Provides a complete set of derivative calculation rules in tables

Corrects the misuse of the traditional product rule in quaternion calculus

Demonstrates applications in signal processing problems

Abstract

A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to handle both analytic and nonanalytic functions. This corrects a flaw in the existing methods, that is, the incorrect use of the traditional product rule. In the framework introduced, the derivatives of quaternion matrix functions can be calculated directly without the differential of this function. Key results are summarized in tables. Several examples show how the quaternion matrix derivatives can be used as an important tool for solving problems related to signal processing.