This paper's main result is to show that under the conditions imposed by the Maloney-Wandell color constancy algorithm, color constancy can in fact be expressed in terms of a simple independent adjustment of the sensor responses - in other words as a von Kries adaptation type of coefficient rule algorithm - so long as the sensor space is first transformed to a new basis. Our overall goal is to present a theoretical analysis connecting many established theories of color constancy. For the case where surface reflectances are 2-dimensional and illuminants are 3-dimensional, we prove that perfect colour constancy can always be solved for by an independent adjustment of sensor responses, which means that the colour constancy transform can be expressed as a diagonal matrix. This result requires a prior transformation of the sensor basis and to support it we show in particular that there exists a transformation of the original sensor basis under which the non-diagonal methods of Maloney-Wandell, Forsyth's MWEXT and Funt and Drew's lightness algorithm all reduce to simpler, diagonal-matrix theories of colour constancy. Our results are strong in the sense that no constraint is placed on the initial sensor spectral sensitivities. In addition to purely theoretical arguments, the paper contains results from simulations of diagonal-matrix-based color constancy in which the spectra of real illuminants and reflectances along with the human cone sensitivity functions are used. The simulations demonstrate that when the cone sensor space is transformed to its new basis in the appropriate manner, a diagonal matrix supports close to optimal colour constancy.