AHL 1.15 (HL)—Eigenvalues and eigenvectors

An eigenvector is a non-zero vector v whose direction a matrix A does not change: Av = λv. The scale factor λ is its eigenvalue.

Find possible λ by solving det(A−λI)=0. For each λ, solve (A−λI)v=0 to obtain the corresponding eigenvectors. The determinant equation identifies scales; the second equation identifies the directions that actually realise them.

For A = [[2,0],[0,3]], the characteristic equation is (2−λ)(3−λ)=0, so λ=2 or 3. The vectors [1,0]^T and [0,1]^T are the matching eigenvectors because A preserves each axis and scales it by its eigenvalue.

An eigenvector cannot be the zero vector, and an arbitrary column of A is not automatically an eigenvector. A repeated eigenvalue may have more than one independent eigenvector, so always solve the null-space equation rather than guessing from the entries.

If a $2\times2$ matrix has two distinct real eigenvalues with independent eigenvectors, put those eigenvectors into columns of $P$ and the eigenvalues into diagonal $D$. Then $A=PDP^{-1}$ and $A^n=PD^nP^{-1}$, where $D^n$ simply raises each eigenvalue to the nth power. This makes repeated population movement or predator–prey transformations interpretable through dominant eigenvalues and their directions.