Backwrd population projection by a generalized inverse

The Leslie population projection matrix may be used to project forward in time the age distribution or age-sex distribution of a population. As it is a singular matrix, it does not have an inverse, and so it is not clear that there is a corresponding procedure for backward projection. In terms of the eigenvalues and eigenvectors of the Leslie matrix, certain generalized inverses are constructed that can sometimes be used advantageously for backward projection.