Eigenvalues and Eigenvectors

Directions That Only Stretch

Most vectors change direction under a linear transformation. But some special vectors are only scaled: the transformation stretches or shrinks them without rotating them. 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 matrix $A$ to eigenvector $v$ gives back $v$ scaled by $\lambda$. Along an eigenvector, complex matrix action becomes simple multiplication by a number.

Repeatedly applying $A$ is governed by its eigenvalues; the largest-magnitude eigenvalue eventually dominates systems like Markov chains.

Finding and Using Eigenvalues

To find eigenvalues, solve the characteristic equation: $$\det(A - \lambda I) = 0$$

This reuses the determinant's singular test because $Av = \lambda v$ means $(A - \lambda I)v = 0$ has a nonzero solution, requiring $A - \lambda I$ to be singular.

Eigenvalues drive major algorithms: PageRank ranks web pages via the dominant eigenvector of a link matrix, and principal component analysis compresses data along top eigenvectors.

Common pitfall: Forgetting that an eigenvector must be nonzero ($v \neq \mathbf{0}$) and must keep its direction. The zero vector trivially satisfies the equation for any $\lambda$, so it is excluded. An eigenvector must map to a scalar multiple of itself, not merely change in size while rotating.