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arXiv · 2110.09282

A Note on Quaternion Skew-Symmetric Matrices

Liqun Qi·Ziyan Luo
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Abstract

The product of a complex skew-symmetric matrix and its conjugate transpose is a positive semi-definite Hermitian matrix with nonnegative eigenvalues, with a property that each distinct positive eigenvalue has even multiplicity. This property plays a key role for Professor Loo-Keng Hua to establish the unitary equivalence theorem for complex skew-symmetric matrices. We show that this property is no longer true for quaternion skew-symmetric matrices. This poses a difficulty for finding the canonical form of a quaternion skew-symmetric matrix under unitary equivalence, which may be crucial for the spectral theory of dual quaternion matrices. We show that the inverse of a quaternion skew-symmetric matrix may not be skew-symmetric though the inverse of a nonsingular complex skew-symmetric matrix is always skew-symmetric. We also present the concept of basic quaternion skew-symmetric matrices.

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Liqun Qi, Ziyan Luo. 2021-10-18. A Note on Quaternion Skew-Symmetric Matrices. https://arxiv.org/abs/2110.09282

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