[Paper Review] Eigenvalues and Singular Values of Dual Quaternion Matrices

Liqun Qi, Ziyan Luo|arXiv (Cornell University)|Nov 24, 2021

Control and Dynamics of Mobile RobotsEngineering6 references20 citations

TL;DR

This paper establishes the spectral theory of dual quaternion matrices, introducing right and left eigenvalues and proving that eigenvalues of dual quaternion Hermitian matrices are dual numbers. It presents a unitary decomposition for such matrices and a singular value decomposition (SVD) for general dual quaternion matrices, showing that singular values are nonnegative dual numbers and defining rank and appreciable rank via positive singular values and their standard parts.

ABSTRACT

The poses of $m$ robotics in $n$ time points may be represented by an $m imes n$ dual quaternion matrix. In this paper, we study the spectral theory of dual quaternion matrices. We introduce right and left eigenvalues for square dual quaternion matrices. If a right eigenvalue is a dual number, then it is also a left eigenvalue. In this case, this dual number is called an eigenvalue of that dual quaternion matrix. We show that the right eigenvalues of a dual quaternion Hermitian matrix are dual numbers. Thus, they are eigenvalues. An $n imes n$ dual quaternion Hermitian matrix is shown to have exactly $n$ eigenvalues. It is positive semidefinite, or positive definite, if and only if all of its eigenvalues are nonnegative, or positive and appreciable, dual numbers, respectively. We present a unitary decomposition of a dual quaternion Hermitian matrix, and the singular value decomposition for a general dual quaternion matrix. The singular values of a dual quaternion matrix are nonnegative dual numbers.

Motivation & Objective

• To develop a comprehensive spectral theory for dual quaternion matrices, which are used to represent robotic poses across time.
• To define and analyze right and left eigenvalues for square dual quaternion matrices, particularly focusing on Hermitian matrices.
• To establish a unitary decomposition for dual quaternion Hermitian matrices and prove they have exactly n eigenvalues.
• To derive a singular value decomposition (SVD) for general dual quaternion matrices and define rank and appreciable rank based on singular values.
• To address foundational gaps in dual quaternion matrix theory, which lacks mature matrix theory compared to real, complex, or quaternion matrices.

Proposed method

• Introduce right and left eigenvalues for square dual quaternion matrices, showing that if a right eigenvalue is a dual number, it is also a left eigenvalue.
• Prove that the standard part of a right eigenvalue of a dual quaternion matrix is a right eigenvalue of the matrix’s standard part.
• Demonstrate that all right eigenvalues of a dual quaternion Hermitian matrix are dual numbers, hence valid eigenvalues, and derive formulas for their standard and infinitesimal parts.
• Establish a unitary decomposition for dual quaternion Hermitian matrices, showing they can be diagonalized and have exactly n eigenvalues.
• Define a 'perfect Hermitian matrix' as a positive semidefinite dual quaternion Hermitian matrix of the form $ A^*A $, which admits a unitary decomposition $ M = U ilde{oldsymbol{ u}}^2 U^* $.
• Construct the SVD of a general $ m \times n $ dual quaternion matrix $ B $ as $ B = \hat{V} \Sigma_t \hat{U}^* $, where $ \Sigma_t $ is a block-diagonal matrix of nonnegative dual numbers.

Experimental results

Research questions

• RQ1 What are the conditions under which a right eigenvalue of a dual quaternion matrix is also a left eigenvalue, and when is it called a true eigenvalue?
• RQ2 Can a dual quaternion Hermitian matrix be unitarily diagonalized, and how many eigenvalues does it possess?
• RQ3 Under what conditions does the matrix $ A^*A $ admit a unitary decomposition $ U \Sigma^2 U^* $, and why is this critical for SVD?
• RQ4 How can the rank and appreciable rank of a dual quaternion matrix be defined and computed using its singular values?
• RQ5 What are the structural and algebraic properties of dual quaternion matrices whose entries are all unit dual quaternions?

Key findings

• An $ n \times n $ dual quaternion Hermitian matrix has exactly $ n $ eigenvalues, all of which are dual numbers.
• A dual quaternion Hermitian matrix is positive semidefinite (or positive definite) if and only if all its eigenvalues are nonnegative (or positive and appreciable) dual numbers.
• The singular values of a dual quaternion matrix are nonnegative dual numbers, and the SVD of a general dual quaternion matrix exists in the form $ B = \hat{V} \Sigma_t \hat{U}^* $, where $ \Sigma_t $ is block-diagonal with nonnegative dual numbers.
• The appreciable rank of a dual quaternion matrix $ B $ equals the rank of its standard part $ B_{\text{st}} $, and the rank of $ B $ is at least the appreciable rank.
• A matrix $ P \in \mathbb{DQ}^{m \times n} $ that is partially unitary (i.e., $ P^*P = I_n $) has both rank and appreciable rank equal to $ n $.
• The matrix $ A^*A $ for any dual quaternion matrix $ A $ is a 'perfect Hermitian matrix', ensuring the existence of a unitary decomposition $ A^*A = U \Sigma^2 U^* $, which enables the SVD construction.