Unitary similarity for dual complex matrices and dual quaternion matrices

Abstract

We study unitary similarity of dual complex matrices and dual quaternion matrices. First, we characterize unitary similarity in terms of the existence of a skew-Hermitian solution to a Sylvester-type matrix equation. For $2\times2$ dual complex matrices, we obtain a complete three-trace criterion when the standard part is nonnormal and analyze the limitations of trace-word criteria when the standard part is normal. We then prove, for every $n\geq1$, that two $n\times n$ dual quaternion matrices are unitarily similar if and only if their dual complex adjoint matrices are unitarily similar, and we extend this result to finite families. As a scalar consequence, we derive a finite Trace-word criterion when the standard part is nonreal and exhibit its failure when the standard part is real.