Some new properties of the right invertible system in control theory

In this paper, we deduce equivalent conditions of controllability, stability and observability of a right invertible system {C,A,B}{C,A,B} by applying the canonical decomposition of {C,A,B}{C,A,B} obtained by Wei et al. [Wei, M., Cheng, X., & Wang, Q. (2010). A canonical decomposition of the right invertible system with applications. SIAM Journal on Matrix Analysis and Applications, 31(4), 1958–1981]. We also derive three equivalent sufficient solvability conditions of the non-regular row-by-row decoupling problem, which are generalization of necessary and sufficient conditions of the regular row-by-row decoupling problem.