The Theory of Quaternion Matrix Derivatives

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.

arXiv commentThis regular paper was submitted to IEEE TSP Jun 03, 2014

References in corpus (7)

• The Complex Gradient Operator and the CR-Calculus
• Extension of Wirtinger's Calculus to Reproducing Kernel Hilbert Spaces and the Complex Kernel LMS
• Quaternion Gradient and Hessian
• Quaternion Involutions
• Quaternion Derivatives: The GHR Calculus
• Fast complexified quaternion Fourier transform
• Properties and Applications of a Restricted HR Gradient Operator

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