This article focuses on the class of linear systems coupled with a nonlinearity satisfying a sector bound or a monotonic condition. First, we present a set of sufficient conditions based on Linear Matrix Inequalities (LMIs) to establish if such a system defines a contraction. Then, we propose a contractive state-feedback control law designed via the use of LMIs which may include nonlinear terms. Furthermore, the design is extended to the case in which an integral action is included in order to achieve constant output regulation. Following a similar approach, the design of contractive observers for the aforementioned class of systems is also considered. Finally, based on previous conditions, we establish a new nonlinear separation principle in which the feedback and the observer gains may be designed independently. Throughout the paper, we illustrate our results by means of practical examples.