Eigenvalues and Singular Values of Dual Quaternion Matrices

Liqun Qi; Ziyan Luo

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arXiv · 2111.12211

Eigenvalues and Singular Values of Dual Quaternion Matrices

Liqun Qi·Ziyan Luo

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Abstract

The poses of $m$ robotics in $n$ time points may be represented by an $m \times n$ 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 $n \times n$ 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.

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Liqun Qi, Ziyan Luo. 2021-11-24. Eigenvalues and Singular Values of Dual Quaternion Matrices. https://arxiv.org/abs/2111.12211

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