Eigenvalues and Eigenvectors

Eigenvalues & Eigenvectors

Think of a transformation as a chaotic warp. Yet, some vectors only stretch—they never rotate. An eigenvector $\vec{v}$ of matrix $A$ is a nonzero vector satisfying $A\vec{v} = \lambda\vec{v}$.

The scalar $\lambda$ is the eigenvalue, scaling that vector. Common pitfall: Eigenvectors only keep their line. Eigenvalues can be negative (flips), zero (collapses), or fractions (shrinks).

Solving & Diagonalisation

Find eigenvalues via the characteristic equation: $\det(A - \lambda I) = 0$. For each root $\lambda_i$, find eigenvectors by solving $(A - \lambda_i I)\vec{v} = \vec{0}$.

Diagonalisation: If $A$ has $n$ independent eigenvectors, $A = PDP^{-1}$, where $D$ holds eigenvalues and $P$ holds eigenvectors as columns. This makes computing $A^k = PD^kP^{-1}$ trivial.

Example: $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ yields $\lambda^2 - 7\lambda + 10 = 0$, giving eigenvalues $5$ and $2$.