Definition:Leslie Matrix

Definition

A Leslie matrix is a square matrix of the form:

$\quad \begin {pmatrix} a_1 & a_2 & \cdots & a_{n - 1} & a_n \\ b_1 & 0 & \cdots & 0 & 0 \\ 0 & b_2 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & b_{n - 1} & 0 \end {pmatrix}$

Also see

• Results about Leslie matrices can be found here.

Source of Name

This entry was named for Patrick Holt Leslie.

Historical Note

The concept of the Leslie matrix was developed by Patrick Holt Leslie in $1945$.

It is used to model population growth, where the $a_i$ describe birth rates, and the $b_i$ survival rates associated with different age groups in a population.

Sources

• 1945: P.H. Leslie: The use of matrices in certain population mathematics (Biometrika Vol. 33, no. 3: pp. 183 – 212)  www.jstor.org/stable/2332297
• 1948: P.H. Leslie: Some further notes on the use of matrices in population mathematics (Biometrika Vol. 35, no. 3: pp. 213 – 245)  www.jstor.org/stable/2332342
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Leslie matrix (P.H. Leslie, 1945)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Leslie matrix (P.H. Leslie, 1945)