Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems

Following an idea originated by Conti for continuous matrix functions, an equivalence relation called summable similarity is defined on pairs of n × n matrix functions A and B. Special cases of the results show that if A and B are summably similar and the system (A) Δxm = Amxm, m = 0, 1, 2, … is uniformly, exponentially, or strictly stable, or has linear asymptotic equilibrium, then the system (B) Δym = Bmym, m = 0, 1, 2,… has the same property. More generally, the same conclusion is obtained under weaker conditions relating A and B.