Block Diagonalization and Multiplicativity of Quaternion $Z$-Block Circulant Matrices with Applications

The motivation of this paper is twofold. We firstly investigate the block-diagonalization of the $z$-block circulant matrix $\mathtt{bcirc_z}(\mathcal A)$, and develop the algorithm $\mathtt{bcirc_z}$-inv for computing the inverse of $\mathtt{bcirc_z}(\mathcal A)$. Second, we establish the multiplicativity of the operator $\mathtt{bcirc_z}$, which yields the equivalence between the QT-product of tensors and the multiplication of their corresponding $z$-block circulant matrices. As applications of these results, we conduct large-scale numerical experiments and analyze the scalability of the Tikhonov-regularized model. We also obtain polar decompositions under the QT-product, develop an algorithm for computing them, and present a large-scale test and a scalability analysis. In addition, we evaluate several rotation strategies based on the QT-polar decomposition through video rotation applications. The results show that the decomposition remains stable and maintains inter-frame consistency while accurately preserving colors.