Generalized Singular Value Decompositions of Dual Quaternion Matrix Triplets
Sitao Ling, Wenxuan Ma, Musheng Wei
TL;DR
This work develops restricted SVD (RSVD) and product-product SVD (PPSVD) for dual-quaternion matrix triplets $(\boldsymbol{A},\boldsymbol{B},\boldsymbol{C})$ to enable structure-preserving decompositions without forming the full product. Building on QSVD/QR foundations for dual-quaternion matrices, it introduces two GSVD-style formulations (DQGSVD) and extends them to triplets through DQRSVD, accompanied by constructive algorithms (compression, triangularization) and existence theorems. A PPSVD framework is then proposed to approximate problems of the form $\min_{\mathrm{rank}(\boldsymbol{X})\le r} \|\boldsymbol{A}(\boldsymbol{B}-\boldsymbol{X})\boldsymbol{C}\|_F$, highlighting that PPSVD reveals appreciable singular values and may require an extra SVD to recover all singular values. The paper includes two illustrative examples demonstrating RSVD and PPSVD for dual-quaternion data, signaling impactful applications in multivariable signal processing and coupled rotation-translation modeling.
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.