Solving a system of quaternion matrix equations by using PSVD for multiple matrices with applications

In this paper, some applications of product singular value decomposition (PSVD) for multiple quaternion matrices are considered. A general system of coupled Sylvester-type quaternion matrix equations with $n$ equations and $n+1$ unknowns is considered by using PSVD for $n$ quaternion matrices, where $n$ is an arbitrary positive integer. Some solvability conditions and general solutions to the system are derived. The general solution to the system is also presented. Some numerical examples are given to illustrate the results of this paper. Moreover, we propose an application to the encryption and decryption of color videos using the system of coupled Sylvester-type quaternion matrix equations. The results demonstrate that the method can encrypt and decrypt video frames effectively, providing high accuracy as validated by metrics such as PSNR, SSIM, and FSIM.