Scattering amplitudes are both a crucial link in the chain from theory to experiment as well as a basic observable: they are computed in theories ranging from the standard model to string theory. One generic lesson learned is that results can be much simpler than their textbook computation suggests. A prime example are the Kawai-Lewellen-Tye (KLT) relations which express the self-scattering of certain gravitational waves in terms of a sum over products of gluon scattering amplitudes, as well as their loop level extension. Simple results beg for a simple explanation, especially in a theory as complicated as general relativity. Here I will show scattering amplitudes can be computed by simply solving all physical constraints, which reduces the problem to linear algebra. Beyond inherent beauty this minimalist approach shows for instance that the KLT relations at tree level are a consequence of gauge invariance, with an important exception in certain effective field theories. Especially exciting is the simplicity of loop level computation in this formalism, for which preliminary one and two loop results in physical gauge and gravity theories will be presented.
