Questions: Applications: Modeling with Linear Algebra

Question 1

A Leslie matrix model of an age-structured fish population is computed and its dominant eigenvalue is found to be λ₁ = 0.94. What does this predict about the long-term population?

• A. The population will grow geometrically toward the stable age distribution given by the dominant eigenvector
• B. The population will decline toward extinction, since λ₁ < 1 means repeated matrix multiplication shrinks the population vector
• C. The population will oscillate, since the dominant eigenvalue is close to 1
• D. Nothing useful — eigenvalues only describe individual-variable behavior, not aggregate population trajectories

Answer: (B)

The dominant eigenvalue λ₁ controls the long-run growth rate when the matrix is applied repeatedly. If λ₁ > 1, the population grows geometrically; if λ₁ = 1, it stabilizes; if λ₁ < 1, it declines toward zero. Here λ₁ = 0.94 < 1 predicts geometric decline. The dominant eigenvector gives the stable age distribution — the proportions each age class approaches — but the magnitude shrinks by a factor of 0.94 per time step. This is the same eigenvalue logic seen abstractly, now giving a concrete biological prediction.

Question 2

The least-squares solution x̂ to an overdetermined system Ax = b is described geometrically as a projection. What is being projected onto what?

• A. The vector x̂ is the projection of x onto the column space of A
• B. The vector b is projected onto the row space of A to find x̂
• C. The product Ax̂ is the projection of b onto the column space of A — the closest achievable output to the observed data
• D. The matrix A is projected onto its own null space to eliminate inconsistency in the system

Answer: (C)

An overdetermined system Ax = b generally has no exact solution because b lies outside the column space of A. The least-squares solution finds x̂ such that Ax̂ is as close as possible to b — meaning Ax̂ is the orthogonal projection of b onto the column space of A. The residual b − Ax̂ is orthogonal to every column of A, which is exactly what the normal equations AᵀAx̂ = Aᵀb capture algebraically. Option A confuses x̂ itself (the parameter vector) with the output Ax̂.

Question 3

The dominant eigenvector of the Leslie matrix for a population model gives the stable age distribution that the population converges to over time, regardless of its initial age structure.

• T. True
• F. False

Answer: True

Any initial population vector can be decomposed as a linear combination of the Leslie matrix's eigenvectors. After repeated multiplication, the component in the direction of the dominant eigenvector (with the largest |λ|) grows faster than all others. Eventually the dominant eigenvector component dominates, so the proportional age distribution converges to it — regardless of starting conditions. The dominant eigenvalue governs the rate at which total population grows or shrinks; the dominant eigenvector governs the shape the population settles into.

Question 4

The PageRank algorithm assigns higher scores to web pages that contain the most high-quality written content, using the dominant eigenvector of a content-quality matrix.

• T. True
• F. False

Answer: False

PageRank uses the link structure of the web — which pages link to which — not the content of pages. The web's link relationships are represented as a matrix, and PageRank computes the dominant eigenvector of that matrix, representing the steady-state distribution of a random web surfer who follows links. Pages linked to by many other important pages receive high scores. This is a purely structural computation; content quality is not analyzed. Being linked to by authoritative sources is the signal of authority.

Question 5

Why do eigenvalues and eigenvectors appear as a unifying theme across applications as different as population biology, web search, and image compression?

Model answer: Eigenvalues and eigenvectors characterize what happens when a linear transformation is applied repeatedly. In any system where a matrix is multiplied against a vector over and over — updating population counts each year, redistributing web traffic through links, extracting dominant patterns from data — the long-run behavior is governed by the dominant eigenvalue and eigenvector. Any initial state can be decomposed into eigenvector components, and after many steps only the component with the largest |λ| survives at scale. This makes eigendecomposition the natural tool for predicting steady states, growth rates, and dominant patterns wherever linear dynamics appear.

The deeper reason eigenvalues unify these applications is that they identify the 'preferred directions' of a linear transformation — the directions that don't rotate, only scale. In a population model, the dominant eigenvector is the age distribution the system naturally evolves toward. In PageRank, it is the natural distribution of web traffic. In SVD, singular values (related to eigenvalues) measure how much of the data lives in each direction. The mathematics is the same; the domains just differ.