Eigenvalues and Eigenvectors

Eigenvector Foundations

Most vectors change direction under a linear transformation. But special nonzero vectors are only scaled, meaning the transformation stretches or shrinks them without rotating.

Such a vector is an eigenvector, and its scale factor is the eigenvalue $\lambda$. The defining equation is: $$A v = \lambda v, \qquad v \neq \mathbf{0}$$

Applying $A$ to eigenvector $v$ returns $v$ scaled by $\lambda$. Eigenvectors act as the natural axes where a transformation operates as pure scaling.

Common pitfall: An eigenvector must be nonzero and keep its line of direction. The zero vector satisfies $Av = \lambda v$ for every $\lambda$, so it is excluded.

Finding Eigenvalues & Eigenspaces

To find eigenvalues, rewrite $Av = \lambda v$ as $(A - \lambda I)v = \mathbf{0}$. For a nonzero solution to exist, the matrix must be singular.

This yields the characteristic equation: $$\det(A - \lambda I) = 0$$

Solving this polynomial gives the eigenvalues. For each eigenvalue, its eigenvectors are the nonzero solutions of $(A - \lambda I)v = \mathbf{0}$, known as the eigenspace.

Long-term behavior: Under repeated application ($A^k$), the largest-magnitude eigenvalue dominates dynamics, powering Markov chains and iterative methods.