Matrix Calculus and Notation

Abstract

In October of 2005, I scribbled in a notebook, "can it possibly be that simple?" I was referring to the sensitivity of transient dynamics (the eventual results appear in Chap. 7), and had just begun to use matrix calculus as a tool. The answer to my question was yes. It can be that simple. This book relies on this set of mathematical techniques. This chapter introduces the basics, which will be used throughout the text. For more information, I recommend four sources in particular. The most complete treatment, but not the easiest starting point, is the book by Magnus and Neudecker (1988). More accessible introductions can be found in the paper by Magnus and Neudecker (1985) and especially the text by Abadir and Magnus (2005). A review paper by Nel (1980) is helpful in placing the Magnus-Neudecker formulation in the context of other attempts at a calculus of matrices. Sensitivity analysis asks how much change in an outcome variable y is caused by a change in some parameter x. At its most basic level, and with some reasonable assumptions about the continuity and differentiability of the functional relationships involved, the solution is given by differential calculus. If y is a function of x, then the derivative dy dx tells how y responds to a change in x, i.e., the sensitivity of y to a change in x. However, the outcomes of a demographic calculation may be scalar-valued (e.g., the population growth rate λ), vector-valued (e.g., the stable stage distribution), or matrix-valued (e.g., the fundamental matrix). Any of these outcomes may be

A demographic and evolutionary analysis of maternal effect senescence

“…We use life table response experiment (LTRE) methods (e.g., refs. 26 and 36 ) to decompose the impact of maternal effect senescence on fitness into contributions from age- and maternal-age-specific survival and fertility. We compare the results for models with and without maternal effect senescence in both high growth rate (laboratory) and low growth rate (simulated natural) environments.…”

Demography and perturbation analyses of the coffee berry borer Hypothenemus hampei (Coleoptera: Curculionidae): Implications for management

“…Prospective analyses are based on the determination of vital rates with highest sensitivities, which can be perturbed in order to decrease the value of population growth rate (λ) to < 1.0. Identification of vital rates with highest contributions to λ can be powerful tools to improve Integrated Pest Management (IPM) programs [ 18 , 35 ].…”

“…Demographic perturbation models are an important tool to understand and manage populations, specifically those models that focus on sensitivity and elasticity analyses [ 17 , 18 ]. These types of analyses can be used in two ways: to protect endangered species or to reduce populations of invasive pests and weeds [ 18 ]. However, most of these analyses have been focused on conservation of endangered populations and less on control of pests.…”

Optimal vaccine allocation for COVID-19 in the Netherlands: A data-driven prioritization

“…Once the matrices are specified, we can evaluate the impact of a single unit of vaccination, as the sensitivity (or elasticity) of the transition matrix (see the general idea of the sensitivity of a matrix in [ 43 ], and its application in infectious disease epidemiology in [ 27 , 44 ]). The change in the number of infections per single vaccination can be formulated as the result of depletion of susceptible and infectious individuals from the population (Eq S4 in S1 Text ), and subsequently, we obtain its effect on the dominant eigenvalue of the next generation matrix that was already introduced in the first step as an approximation with observed data.…”