01 · Lesson 07

Eigenvalues and eigenvectors

A matrix usually changes both the size and direction of a vector. An eigenvector is a special direction that the matrix does not rotate away from; the corresponding eigenvalue tells us how that direction is scaled.

Scenario: long-term population structure

A juvenile–adult population is updated each year by a fixed projection matrix. Starting proportions may differ, but after many years the ratio of juveniles to adults often approaches a stable pattern while the total population grows or declines by an approximately constant factor.

This pattern is explained by a dominant eigenvalue and its eigenvector.

Definition

For a square matrix \(A\), a non-zero vector \(\mathbf v\) is an eigenvector if

\[A\mathbf v=\lambda\mathbf v.\]

The scalar \(\lambda\) is the corresponding eigenvalue. The transformation may stretch, shrink or reverse the eigenvector, but the result remains on the same line.

The zero vector is excluded because \(A\mathbf 0=\lambda\mathbf 0\) would hold for every \(\lambda\) and would provide no special direction.

Finding eigenvalues

Rearranging gives

\[(A-\lambda I)\mathbf v=\mathbf 0.\]

A non-zero solution exists only when \(A-\lambda I\) is singular. Therefore eigenvalues satisfy the characteristic equation

\[\det(A-\lambda I)=0.\]

After finding an eigenvalue, solve \((A-\lambda I)\mathbf v=0\) for its eigenvectors. Multiplying an eigenvector by any non-zero scalar gives another eigenvector for the same eigenvalue.

A simple example

For

\[A=\begin{pmatrix}2&0\\0&0.5\end{pmatrix},\]

the coordinate directions are eigenvectors:

\[A\begin{pmatrix}1\\0\end{pmatrix}=2\begin{pmatrix}1\\0\end{pmatrix},\qquad A\begin{pmatrix}0\\1\end{pmatrix}=0.5\begin{pmatrix}0\\1\end{pmatrix}.\]

Repeated updating amplifies the first component by \(2^k\) and reduces the second by \((0.5)^k\). Unless the initial first component is zero, the first direction eventually dominates.

Dominant eigenvalue in population models

For a non-negative stage-projection matrix under suitable connectivity conditions, the eigenvalue with largest magnitude determines long-term multiplication, and a corresponding positive eigenvector determines stable stage proportions.

Dominant eigenvalue	Long-term interpretation per step
\(\lambda_1>1\)	Population grows
\(\lambda_1=1\)	Population size remains asymptotically constant
\(0<\lambda_1<1\)	Population declines

The eigenvector gives proportions only after normalisation, for example by dividing its components by their sum.

Continuous-time stability

Near an equilibrium of an ODE system, eigenvalues of the Jacobian matrix describe local behaviour. For continuous-time systems:

all real parts negative usually indicate local asymptotic stability;
at least one positive real part indicates instability;
non-zero imaginary parts can produce oscillatory behaviour.

This differs from a discrete-time update, where local stability requires eigenvalue magnitudes below one. The model’s time structure determines the correct criterion.

What eigenvalues do not guarantee

A dominant pattern may require time to emerge and can depend on the initial state.
Repeated or complex eigenvalues require careful interpretation.
Local linear stability does not automatically describe distant nonlinear behaviour.
A biologically unrealistic matrix can still have mathematically valid eigenvalues.
Numerically close eigenvalues may make convergence slow and estimates sensitive.

What this lesson adds

You can now define eigenvalues and eigenvectors, understand the characteristic equation, interpret dominant population growth and stable structure, and distinguish continuous-time from discrete-time stability criteria.