The hyperbolic CS decomposition of tensors based on the C-product

[EN] This paper studies the issues about the hyperbolic CS decomposition of tensors under the C-product. The aim of this paper is fourfold. Firstly, we establish the CS decomposition of a complex unitary tensor, including the thin version and the standard version. The corresponding numerical algorithm is also given. Next, we define three kinds of tensors, i.e., the strong unitary tensor, the mode-1 strong unitary tensor, the mode-2 strong unitary tensor, and we give the CS decomposition of the last two kind of tensors aforementioned. Numerical algorithms are also obtained to compute the two types of the CS decompositions. Moreover, we give the definition of another three classes of tensors, called the J $J$ -orthogonal tensor, the mode-1 strong J $J$ -orthogonal tensor and the mode-2 strong J $J$ -orthogonal tensor. The corresponding hyperbolic CS decompositions and numerical algorithms are also established. Finally, we give an application to the computation of the C-eigenvalues of the orthogonal tensor. Numerical examples are given to test our results.