[Paper Review] Minimax principle for right eigenvalues of dual quaternion matrices and their generalized inverses

Ling Chen, Liqun Qi+1 more|arXiv (Cornell University)|Mar 7, 2022

Optical measurement and interference techniques4 citations

TL;DR

This paper establishes a minimax principle for right eigenvalues of dual quaternion Hermitian matrices and develops generalized inverses for dual quaternion matrices using singular value decomposition and a newly derived Cauchy-Schwarz inequality. The key contribution is a complete characterization of {1}, {3}, and {4}-type generalized inverses via structured matrix conditions, enabling spectral and inverse analysis in 3D motion modeling and multi-agent control systems.

ABSTRACT

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency for a set of dual quaternion vectors, and study some related basic properties for the set of dual quaternion vectors and dual quaternion matrices. We present a minimax principle for right eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an important inequality for singular values of dual quaternion matrices. We finally introduce the concept of generalized inverse of dual quaternion matrices, and present the necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.

Motivation & Objective

• To formalize three distinct concepts of right linear independence for dual quaternion vectors, enabling structural analysis of dual quaternion matrix systems.
• To establish a minimax principle for eigenvalues of dual quaternion Hermitian matrices, extending classical spectral theory to non-commutative algebras.
• To derive a dual quaternion matrix version of the Fan-Hoffman inequality using singular value decomposition and a new Cauchy-Schwarz inequality.
• To define and characterize Moore-Penrose-type generalized inverses for dual quaternion matrices, particularly for {1}, {3}, and {4}-inverses.
• To provide necessary and sufficient conditions for a matrix to be a generalized inverse of another dual quaternion matrix, grounded in structural matrix decomposition.

Proposed method

• Introduces three definitions of right linear independence for dual quaternion vectors, analyzing their implications on matrix rank and vector space structure.
• Derives a new Cauchy-Schwarz inequality for dual quaternion vectors, essential for bounding inner products and singular values.
• Applies singular value decomposition (SVD) of dual quaternion matrices to establish a Fan-Hoffman-type inequality for singular values.
• Proposes a parametric form for generalized inverses using the split matrix decomposition $ A = A_{\rm st} + A_{\rm I}\epsilon $, where $ A_{\rm st} $ and $ A_{\rm I} $ are real matrices.
• Derives necessary and sufficient conditions for $ X = A_{\rm st}^\dagger - A_{\rm st}^\dagger A_{\rm I} A_{\rm st}^\dagger \epsilon $ to be a {1}, {3}, or {4}-inverse via matrix equations involving $ A_{\rm st}^\dagger $ and Hermitian conditions.
• Validates the existence and structure of Moore-Penrose generalized inverses when all nonzero singular values are appreciable, using explicit SVD-based inversion formula.

Experimental results

Research questions

• RQ1 How can right linear independence be consistently defined for sets of dual quaternion vectors, and what are the implications for matrix rank and vector space structure?
• RQ2 What is the minimax characterization of eigenvalues for dual quaternion Hermitian matrices, and how does it extend classical variational principles?
• RQ3 Can a Fan-Hoffman-type inequality be established for singular values of dual quaternion matrices, and what is its form?
• RQ4 What are the necessary and sufficient conditions for a dual quaternion matrix to be a {1}, {3}, or {4}-generalized inverse of another?
• RQ5 Under what conditions does a Moore-Penrose generalized inverse exist for a dual quaternion matrix, and what is its explicit form?

Key findings

• A minimax principle for eigenvalues of dual quaternion Hermitian matrices is established, generalizing the Rayleigh-Ritz variational principle to the dual quaternion setting.
• A new Cauchy-Schwarz inequality for dual quaternion vectors is derived, enabling the analysis of inner products and singular values in non-commutative settings.
• An inequality for singular values of dual quaternion matrices is proven using SVD and the new Cauchy-Schwarz inequality, analogous to the Fan-Hoffman inequality for complex matrices.
• Necessary and sufficient conditions for a matrix $ X $ to be a {1}-generalized inverse of $ A $ are given by $ (I_m - A_{\rm st}A_{\rm st}^\dagger)A_{\rm I}(I_n - A_{\rm st}^\dagger A_{\rm st}) = O $, ensuring the standard inverse condition.
• For $ X $ to be a {3}-generalized inverse, the condition $ (I_m - A_{\rm st}A_{\rm st}^\dagger)A_{\rm I}A_{\rm st}^\dagger = (A_{\rm st}^*)^\dagger A_{\rm I}^*(I_m - A_{\rm st}A_{\rm st}^\dagger) $ must hold, ensuring $ AX $ is Hermitian.
• For $ X $ to be a {4}-generalized inverse, the condition $ A_{\rm st}^\dagger A_{\rm I}(I_n - A_{\rm st}^\dagger A_{\rm st}) = (I_n - A_{\rm st}^\dagger A_{\rm st}) A_{\rm I}^* (A_{\rm st}^*)^\dagger $ must be satisfied, ensuring $ XA $ is Hermitian.