P. H. Leslie. 1945.

P. H. Leslie. 1945. On the Use of Matrices in Certain Population Mathematics. Biometrika 33(3): 183-212. Oxford University Press.

Leslie's work, rather than that of his predecessors Bernardelli and Lewis, is most commonly cited in the widespread literature using matrices, largely for the reason that Leslie worked out the mathematics and the application with great thoroughness. Some of his elaboration was designed to save arithmetic-for example his transformation of the projection matrix into an equivalent form with unity in the subdiagonal positions. Such devices, like a considerable part of classical numerical analysis, are unnecessary in a computer era.

The case of double roots seemed to require attention, for Leslie could not then know that no case of double roots would ever arise with real data. In the present excerpt we omit this, as well as the spectral decomposition of the matrix, which has not found extensive application.

Among the latent roots or eigen values and the corresponding stable vectors, as Leslie points out, demographic and biological interest is confined to three. The first, of largest absolute value, is positive and represents the ratio of population at the end of a cycle to that at the beginning, when the process has been operating for a considerable period; it provides the component of geometric increase. The second and third roots produce waves of diminishing amplitude having the length of the generation, usually 25 to 30 years. These waves measure the echo effect-after a baby boom they provide for a smaller boom a generation later-on the condition that the age·specific rates of birth and death remain constant. D. P. Smith et al., Mathematical Demography
© Springer-Verlag Berlin · Heidelberg 1977

1. . Introduction 183
2. . Derivation of the matrix elements 184
3. . Numerical example 185
4. . Properties of the basic matrix 187
5. . Transformation of the co-ordinate system 188
6. . Relation between the canonical1
7. . Properties of the stable vectors . 193
8. . The spectral set of operators 194
9. . Reduction of B to classical canonical form 195
10. . The relation between rp and l/r vectors 197
11. . Case of repeated latent roots 198
12. . The approach to the stable age distribution 199
13. . Special case of the matrix with only a single non-zero F. element 200
14. . Numerical comparison with the usual methods of computation 201
15. . Further practical applications 207

Appendix:

(1) The tables of mortality and fertility 209

(2) Calculation of the rate of increase 210

(3) Numerical values of the matrix elements 212

References 212