Generalized singular value decompositions of dual quaternion matrices and beyond
Sitao Ling, Wenxuan Ma, Musheng Wei
TL;DR
The paper addresses extending generalized singular value decomposition to dual-quaternion matrices, building on prior dual-quaternion SVD results. It develops two quotient-type DQGSVD forms for dual-quaternion matrix pairs with equal numbers of columns, plus product-type DQPSVD and canonical correlation DQCCD decompositions, and it establishes QR and CS decompositions tailored to the dual-quaternion setting. Central to the contribution is handling the dual-part structure, which yields distinct forms from real, complex, or quaternion GSVDs and requires careful construction of unitary dual-quaternion factors and nonsingular X-like blocks. The work lays a theoretical foundation with potential applications to dual-quaternion matrix equations and hand-eye calibration, offering multiple decompositions suited to different problem configurations and numerical considerations.
Abstract
In high-dimensional data processing and data analysis related to dual quaternion statistics, generalized singular value decomposition (GSVD) of a dual quaternion matrix pair is an essential numerical linear algebra tool for an elegant problem formulation and numerical implementation. In this paper, building upon the existing singular value decomposition (SVD) of a dual quaternion matrix, we put forward several types of GSVD of dual quaternion data matrices in accordance with their dimensions. Explicitly, for a given dual quaternion matrix pair $\{{\boldsymbol A}, {\boldsymbol B}\}$, if ${\boldsymbol A}$ and ${\boldsymbol B}$ have the same number of columns, we investigate two forms of their quotient-type SVD (DQGSVD) through different strategies, which can be selected to use in different scenarios. Three artificial examples are presented to illustrate the principle of the DQGSVD. Alternatively, if ${\boldsymbol A}$ and ${\boldsymbol B}$ have the same number of rows, we consider their canonical correlation decomposition (DQCCD). If ${\boldsymbol A}$ and ${\boldsymbol B}$ are consistent for dual quaternion matrix multiplication, we present their product-type SVD (DQPSVD). As a preparation, we also study the QR decomposition of a dual quaternion matrix based on the dual quaternion Householder transformation, and introduce the CS decomposition of an 2-by-2 blocked unitary dual quaternion matrix. Due to the peculiarity of containing dual part for dual quaternion matrices, the obtained series of GSVD of dual quaternion matrices dramatically distinguish from those in the real number field, the complex number field, and even the quaternion ring, but can be treated as an extension of them to some extent.