The Lorentz boost transforms 4 vectors from one intertial reference frames to the other. The common form of this transformation is
where , , is the relative velocity between the two reference frames and is the speed of light. We would like to generalise this transformation to a general velocity vector . To so this, we split the position vector into a component parallel and perpendicular to . The parallel component transforms in the way described above, and the perpendicular component does not change. Hence
Hence the transformation matrix can be written in the following form
where is an outer an outer product (i.e. a tensor) rather than a scalar.