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Unitary rational functions: The scaled quaternion case

Daniel Alpay, Ilwoo Cho, Mihaela Vajiac

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.

Unitary rational functions: The scaled quaternion case

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.

Paper Structure

This paper contains 29 sections, 56 theorems, 344 equations.

Table of Contents

  1. Introduction
  2. The rings $\mathbb H_t$
  3. Definition and some lemmas
  4. Point evaluations
  5. The adjoint $\circledast$
  6. Convergence in $\mathbb H_t^n$
  7. $\circledast$-symmetric matrices
  8. $\circledast$-Positive matrices
  9. $\circledast$-positive definite kernels
  10. Realization theory in $\mathbb H_t$
  11. Definitions
  12. Observability
  13. Controllability
  14. Minimal realizations
  15. Minimal factorizations
  16. ...and 14 more sections

Key Result

Theorem 1.4

Let $J\in\mathbb C^{n\times n}$ be a signature matrix, and let $R(z)$ be a $\mathbb C^{n\times n}$-valued rational function regular at the origin, and with realization mini-real-1, $R(z)=D+zC(I_N-zA)^{-1}B$, assumed minimal. Then the following are equivalent: $(1)$$R$ takes $J$-unitary-values on the $(2)$ It holds and there exists an Hermitian invertible matrix $H$, uniquely determined from the g

Theorems & Definitions (133)

  • Remark 1.1
  • Remark 1.2
  • Example 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 123 more