Decoupling a System of Linear Differential Equations into Blocks

Various problems on reducing a system of linear differential equations to several independent lower-order subsystems are overviewed. The construction of auxiliary square matrices for which similarity transformation is applied is substantiated. This allows the results obtained for the problem of reducing two square matrices to be transferred to different classes of problems: obtaining the maximum possible number of blocks, proof of the uniqueness of the obtained decoupling, revealing the hidden symmetry of the system under study, detecting the loosely coupled subsystems, and creating efficient computational decoupling algorithms.