Eigenvalues and Eigenvectors

The Directions That Don't Rotate

Eigenvalues answer the most fundamental question about a linear system: does it grow, shrink, or stay the same, and in which directions?

Given a square matrix $A$, an eigenvector $\mathbf{v}$ is a non-zero vector that $A$ merely scales rather than rotates:

$$A\mathbf{v} = \lambda \mathbf{v}$$

The scalar $\lambda$ is the eigenvalue, representing the stretch, compression, or flip factor.

To find them, rearrange as $(A - \lambda I)\mathbf{v} = \mathbf{0}$. For a non-zero solution, the matrix must be singular:

$$\det(A - \lambda I) = 0$$

This is the characteristic equation, a polynomial whose roots are the eigenvalues.

The $2 \times 2$ Shortcut

For any $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, skip full expansion using the trace ($a+d$) and determinant ($ad-bc$).

Economic and Data Applications:

• Input-output models: Dominant eigenvalue shows if an economy yields a net surplus.
• Dynamic systems: Stability requires all $|\lambda| < 1$; otherwise, the system explodes.
• Markov chains: The stationary distribution matches $\lambda = 1$.
• PCA: Eigenvectors of the covariance matrix reveal directions of greatest variation.