Generalized singular value decompositions of dual quaternion matrices and beyond

TL;DR

This paper develops various types of generalized singular value decompositions for dual quaternion matrices, extending classical linear algebra tools to high-dimensional dual quaternion data analysis with new algorithms and theoretical insights.

Contribution

It introduces multiple GSVD forms for dual quaternion matrices, including quotient-type, canonical correlation, and product-type SVDs, along with QR and CS decompositions, expanding the mathematical framework for dual quaternion data.

Findings

Proposes quotient-type SVD (DQGSVD) for same-column matrices.

Introduces canonical correlation decomposition (DQCCD) for same-row matrices.

Develops product-type SVD (DQPSVD) for compatible matrix products.

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.