Quadratic multidimensional functions play a very important role in the understanding of general nonlinear functions. Convexity of quadratic functions is linked in a natural way from its geometrical definition all the way to the properties of its matrix eigenspectrum.Indeed, to second order expansion, and close to the expansion point, any nonlinear function can be approximated by a quadratic – thus providing a crucial link and understanding of the local behaviour and convexity properties of general functions.