Diagonalization of skew-Hermitian dual quaternion matrices and $\phi$-Hermitian dual quaternion matrices

In this article, we present the diagonalization of skew-Hermitian dual quaternion matrices in the sense of unitary equivalence. For a unit quaternion $\phi$, we also obtain the diagonalization result of $\phi$-Hermitian dual quaternion matrices in the sense of unitary $\phi$-congruence, which generalizes the Autonne–Takagi factorization of symmetric complex matrices and $\phi$-Hermitian quaternion matrices.