Generalized Singular Value Decompositions of Dual Quaternion Matrix Triplets

TL;DR

This paper introduces two new types of generalized singular value decompositions for dual quaternion matrix triplets, enabling advanced matrix factorizations for coupled rotation-translation signals in engineering.

Contribution

It develops the restricted and product-product SVDs for dual quaternion matrices, extending standard SVD concepts to handle dual quaternion triplets with distinct inner products.

Findings

Two types of GSVDs are formulated for dual quaternion matrices.

Illustrative examples demonstrate the feasibility of the proposed decompositions.

The methods facilitate efficient processing of rotation-translation signals.

Abstract

In signal processing and identification, generalized singular value decomposition (GSVD), related to a sequence of matrices in product/quotient form are essential numerical linear algebra tools. On behalf of the growing demand for efficient processing of coupled rotation-translation signals in modern engineering, we introduce the restricted SVD of a dual quaternion matrix triplet $(\boldsymbol{A},\boldsymbol{B},\boldsymbol{C})$ with $\boldsymbol{A}\in {\bf \mathbb{DQ}}^{m \times n}$, $\boldsymbol{B} \in {\bf \mathbb{DQ}}^{m \times p}$, $\boldsymbol{C} \in {\bf \mathbb{DQ}}^{q\times n}$, and the product-product SVD of a dual quaternion matrix triplet $(\boldsymbol{A},\boldsymbol{B},\boldsymbol{C})$ with $\boldsymbol{A}\in {\bf \mathbb{DQ}}^{m \times n}$, $\boldsymbol{B} \in {\bf \mathbb{DQ}}^{n \times p}$, $\boldsymbol{C} \in {\bf \mathbb{DQ}}^{p\times q}$. The two types of GSVDs represent a sophisticated matrix factorization that accounts for a given dual quaternion matrix in conjunction with two additional dual quaternion matrices. The decompositions can be conceptualized as an adaptation of the standard SVD, where the distinctive feature lies in the application of distinct inner products to the row and column spaces. Two examples are outlined to illustrate the feasibility of the decompositions.