Unitary rational functions: The scaled quaternion case

TL;DR

This paper extends the theory of rational functions to the scaled quaternion setting, developing minimal realizations, factorizations, and unitarity concepts in this non-commutative algebraic context.

Contribution

It introduces a novel framework for rational functions over scaled quaternions, including new definitions of adjoint and unitarity, and develops associated realization and factorization theories.

Findings

Established minimal realization theory for scaled quaternion rational functions

Defined a new notion of unitarity in the scaled quaternion context

Developed initial theory of matrices over the scaled quaternion ring

Abstract

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternionic settings, and in particular, the adjoint is not the classical adjoint and we use properties of real Hilbert spaces. This adjoint allows to define the counterpart of unitarity for matrix-rational functions, and we develop the corresponding theories of realizations and unitary factorizations. We also begin a theory of matrices in the underlying rings.