On the Research of the Secondary Transpose of Tensors via the C-Product

This paper develops a secondary transpose for third-order real tensors under the C-product. The operation is defined in the discrete-cosine-transform domain, reverses the order of the C-product, and is naturally related to the usual transpose through secondary identity tensors. We establish its basic algebraic properties, including involution, product reversal, and the decomposition into secondary symmetric and secondary antisymmetric parts. We then introduce strongly secondary orthogonal tensors and Φ-secondary orthogonal tensors, and obtain equivalent characterizations, closure properties, multideterminant identities, and spectral consequences. Finally, we apply the framework to C-product linear systems and to the secondary Moore–Penrose inverse, including a sufficient-condition representation and a worked DCT-domain example.