One of the latest and potentially most fruitful methods of population projection is to apply matrix algebra. Essentially, these methods follow the logic of the cohort-survival technique. The initial age-and-sex distribution is similarly represented as column sector but the incidence of births and deaths is handled by means of a ‘survivorship matrix’ which operates on the original population (column sector) to age the population through successive time periods, simultaneously performing the calculations of births and deaths. This matrix operator is of the form shown in Fig. 4. The example given here is of for the female population grouped into quinary ranges of age (0-4, 5-9, 10-14, …85 and over). b4, b5,…..b10 are the age-specific birth rates and the sub-diagonal terms, s1, s2, s3, ….s17 the probabilities or survival from the nth to n+1th quinary age-range.
The effects of migration may be introduced by replacing the initial population vector by a matrix in which columns are regions or areas and where rows, as before, are the age groups. Similarly the survivorship matrix is replaced by a set of matrices, one for each region.
Migration is then introduced by means of a set of transition matrices (one for each age group) in which the elements represent the probability that an individual of that age group and in region i at any time will move to region j in the next time period.
The population projection is then carried out by multiplying the initial population matrix (columns are regions, rows are age-groups) by survivorship matrix and then adding the net migration matrix (whose rows are age groups and whose columns are regions).
The method is appealing in its elegance and yet leaves certain questions unanswered for example survival-and-birth-rates are assumed constant over time and over each age group. None of these assumptions could easily be defended in the majority of practical applications. Nevertheless, the method appears to have enormous potential, especially as it seems capable of handling interregional migrations, both inward and outward, and specific to age group and sex. It also seems capable of doing so with economy and elegance of operation in a manner ideally suited to simple computer operations.