The Theory of Quaternion Matrix Derivatives

Dongpo Xu, Danilo P. Mandic

Abstract: A systematic framework for the calculation of the derivatives of quaternion matrix functions with respect to quaternion matrix variables is introduced. The proposed approach is equipped with the matrix product and chain rules and applies to both analytic and nonanalytic functions of quaternion variables. This rectifies a mathematical shortcut in the existing methods, which incorrectly use the traditional product rule. We also show that within the proposed framework, the derivatives of quaternion matrix functions can be calculated directly, without using quaternion differentials or resorting to the isomorphism with real vectors. Illustrative examples show how the proposed quaternion matrix derivatives can be used as an important tool for solving optimization problems in signal processing applications.