Eigenvalues and Eigenvectors

Directions a Transformation Preserves

Most vectors change direction under a linear transformation. But some special nonzero vectors are only scaled — 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: $$A v = \lambda v, \qquad v \neq \mathbf{0}.$$ Applying $A$ to its eigenvector $v$ returns $v$ scaled by $\lambda$ — same direction (or exactly reversed), new length. Eigenvectors are the "natural axes" along which a complicated transformation acts simply, as pure scaling.

To find eigenvalues, rewrite $Av = \lambda v$ as $(A - \lambda I)v = \mathbf{0}$. For a nonzero $v$ to exist, the matrix $A - \lambda I$ must be singular, i.e. have zero determinant. This gives the characteristic equation: $$\det(A - \lambda I) = 0,$$ a polynomial in $\lambda$ (the characteristic polynomial) whose roots are the eigenvalues. Once an eigenvalue $\lambda$ is known, its eigenvectors are the nonzero solutions of $(A - \lambda I)v = \mathbf{0}$ — the null space of $A - \lambda I$, called the eigenspace.

Eigenvalues reveal a transformation's behavior:

• Magnitude $> 1$ stretches that direction; $< 1$ shrinks it; negative also flips it.
• Under repeated application ($A^k$), the largest-magnitude eigenvalue dominates the long-term behavior — the key to dynamical systems, Markov chains, and iterative methods.
• The eigenvalues of a matrix sum to its trace and multiply to its determinant — handy consistency checks.

Common pitfall: forgetting that an eigenvector must be nonzero and keeps its direction (only length changes). The zero vector satisfies $A\mathbf{0} = \lambda\mathbf{0}$ for every $\lambda$, so it is excluded by definition. And an eigenvector is not merely a vector whose length the matrix changes — it must map to a scalar multiple of itself (same or exactly opposite direction), not one that rotates.