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 beautifully simple: $$A v = \lambda v, \qquad v \neq \mathbf{0}.$$ Applying the matrix $A$ to its eigenvector $v$ gives back $v$ scaled by $\lambda$ — same direction, new length.

Eigenvalues reveal the "natural axes" of a transformation. Along an eigenvector, the complicated matrix action becomes simple multiplication by a number. This is why they matter:

• An eigenvalue of magnitude $> 1$ means that direction is stretched; magnitude $< 1$ means shrunk; a negative eigenvalue also flips the direction.
• Repeatedly applying $A$ (as in a dynamical system or a Markov chain) is governed by the eigenvalues — the largest-magnitude one eventually dominates.

To find eigenvalues you solve the characteristic equation $\det(A - \lambda I) = 0$ — reusing the determinant's "singular" test, because $Av = \lambda v$ means $(A - \lambda I)v = 0$ has a nonzero solution, which requires $A - \lambda I$ to be singular.

Eigenvalues are among computing's most applied ideas: PageRank ranks web pages via the dominant eigenvector of a link matrix; principal component analysis compresses data along top eigenvectors; and stability of systems, vibration modes, and quantum states are all eigenvalue problems.

Common pitfall: forgetting that an eigenvector must be nonzero and that it keeps its direction (only the length changes). The zero vector trivially satisfies $Av = \lambda v$ for any $\lambda$, so it is excluded by definition. And an eigenvector is not just any vector the matrix scales the length of — it must map to a scalar multiple of itself (same or exactly opposite direction), not merely change in size while rotating.