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This paper derives a novel approach to prove contraction of nonlinear dynamical systems, based on the use of non-Euclidean norms and their associated matrix measures. A graphical procedure is developed to derive conditions for a system to be contracting. Such conditions can also be used to design control strategies to make a system contracting, or to design consensus and synchronization strategies for networks of nonlinear oscillators. After presenting the main steps of the approach and its proof, both for continuous-time and discrete-time systems, we illustrate the theoretical derivations on a set of representative examples.