Abstract: The inverse eigenvalue problem of quaternion self-conjugated symplectic tensors under Einstein product is proposed and discussed. Firstly, the properties and characteristic structures of conjugate symplectic tensors are obtained by using the transformation operator of quaternion tensors. Secondly, for the given ${I_1}{I_2} \cdots {I_N}$ characteristic pairs of quaternion tensors, the quaternion self-conjugated symplectic tensor $\mathcal{S}$ is found to contain all the given characteristic pairs. As an application, the necessary and sufficient conditions for the existence of conjugate symplectic tensor solutions of the quaternion tensor equation $\mathcal{B}\ast_{N}\mathcal{S} = \mathcal{D}$ and the expressions of their solutions are given, and the feasibility of the proposed method is verified by numerical examples.

Key words: Quaternion, Einstein product, Conjugate symplectic tensor, Characteristics structure, Inverse eigenvalue problem