On bounded matrices and kinematic similarity
C. E. Langenhop · Transactions of the American Mathematical Society · 1960
has discussed the concept of kinematic similarity of matrices and Lillo [3] has used this idea in the study of almost periodic solutions of differential equations.The concept as considered by Markus and others (see [4] for further references) evidently has its origins in Liapounoff's work [2, where the case with which we are concerned in the present paper is referred to as a "reducible system of equations."Markus restricted his considerations to matrix functions of a real variable t which are bounded on a half-line, say 0 ^t< », but almost periodic matrices are bounded on the whole real line.Accordingly we find it more natural, as it would be in the applications Lillo makes, to define Mn as the set of all n by n matrices whose entries are complex-valued functions of a real variable / which are continuous and bounded on the whole real line.Let A(t), B(t)