How Are Eigenvalues Used to Find Powers of a Matrix (Population Models; Predator-Prey Models)?

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This is AHL 1.15: Eigenvalues and eigenvectors, so it is HL-only content.

The Mechanism

For a diagonalizable matrix $M$:

1. Find its eigenvalues from $\det(M-\lambda I)=0$.
2. Find an eigenvector corresponding to each eigenvalue.
3. Form $P$ using the eigenvectors as columns.
4. Form $D$ with the corresponding eigenvalues on its diagonal.
5. Use

If $D=\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix}$, then

The method is most useful when $n$ is large, because direct multiplication repeats the same matrix product many times. For the IB syllabus, diagonalization is restricted to $2\times2$ matrices with distinct real eigenvalues, ensuring two independent eigenvectors can form a valid $P$.

Diagonalization changes coordinates to the eigenvector directions. In those directions, applying $M$ simply multiplies each component by its eigenvalue, so applying $M$ repeatedly multiplies that component by $\lambda^n$. This explains both the calculation and the model's long-term behaviour.

Population Example

Suppose movement between two populations is modelled by

Its eigenvalues are $1$ and $0.5$, with corresponding eigenvectors $\begin{pmatrix}3\\2\end{pmatrix}$ and $\begin{pmatrix}1\\-1\end{pmatrix}$. If $\mathbf{p}_0=\begin{pmatrix}100\\0\end{pmatrix}$, then

Therefore,

This decomposition separates the stable population pattern from the temporary imbalance. As $n\to\infty$, $(0.5)^n\to0$, so $\mathbf{p}_n\to\begin{pmatrix}60\\40\end{pmatrix}$. The eigenvalue $1$ determines the steady-state distribution, while $0.5$ controls how quickly it is approached.

In predator-prey models, components along different eigenvectors can represent persistent, growing, or declining population patterns. Negative or complex eigenvalues may also indicate alternating or oscillatory behaviour, although interpretation depends on the model and time step.

Exam Technique

In an HL Paper 1 or Paper 2 response, show $M=PDP^{-1}$ before writing $M^n=PD^nP^{-1}$. Keep eigenvalues and eigenvectors in matching order, retain full GDC precision, and interpret the long-term result in context rather than giving only a matrix.