We study linear systems in which both the coefficients of the linear combinations and the variables which are combined linearly are only partially known. This includes the two logical extremes completely known and completely unknown. The systems we study include the usual linear systems of simultaneous equations, as well as
multivariate analysis systems. Throughout, we use unweighted least squares loss functions and majorization algorithms to minimize them. We incorporate both errors-in-equations and errors-in-variables as additional unknowns into the loss function, and arrive at algorithms for decompositions of partially unknown matrices.