The recently introduced generalized Hamiltonian–Real (GHR)
calculus comprises, for the first time, the product and chain
rules that makes it a powerful tool for quaternion-based
optimization and adaptive signal processing. In this paper,
we introduce novel dual relationships between the GHR
calculus and multivariate real calculus, in order to provide a
new, simpler proof of the GHR derivative rules. This further
reinforces the theoretical foundation of the GHR calculus and
provides a convenient methodology for generic extensions of
real- and complex-valued learning algorithms to the quaternion
domain.