Mathematics: Applications and Interpretation · Topic 1: Number and algebra

AHL1.15 — Eigenvalues and eigenvectors

Mathematics: Applications and Interpretation · SL / HL · syllabus-mapped notes

1. AIAHL1.15.1

   Eigenvalues and the characteristic polynomial

   $A\mathbf{v}=\lambda\mathbf{v}$; solve $\det(A-\lambda I)=0$ for the eigenvalues.

2. AIAHL1.15.2

   Finding eigenvectors

   For each $\lambda$, solve $(A-\lambda I)\mathbf{v}=\mathbf{0}$.

3. AIAHL1.15.3

   Diagonalization and powers of a matrix

   $A=PDP^{-1}$ and $A^{n}=PD^{n}P^{-1}$ with distinct real eigenvalues.

4. AIAHL1.15.4

   Applications of eigenvalues

   Long-run behaviour: population movement, predator/prey, invariant states.