Full- and Minimum-Order Quaternionic MIMO Observers: Spectral Assignment and Noise Geometry

State observation over the Hamilton division algebra is formulated as a structured realization problem for quaternionic multi-input multi-output systems. A left-intertwiner equation unifies companion assignment, row-oriented robust pole assignment, and coupled target realization with explicit noncommutative product order. A unitary measured-subspace decomposition yields a derivative-free observer with n-r dynamic quaternion states, where r = rank C, and reduces observability and detectability to a smaller pair. This order is minimal within the stated quaternion-linear full-state observer class. For full-order observers, structured H2 Lyapunov equations and an exact fixed-target gradient allow noise attenuation without moving the spectrum. For minimum-order observers, the reconstruction exposes the direct noise term and its interaction with pole-invariant injection freedom. Discrete-time formulas include the finite contribution of direct feedthrough, and correlated redundant outputs are characterized as noise references. Reproducible studies verify the gradients, compare quaternion, structured-complex, and real representations, expose the noise-cost nonuniqueness of maximum-volume designs, and test coupled repeated-class targets and a multi-output minimum-order observer.