Eigenvalues and Eigenvectors

The Directions That Don't Rotate

Given a square matrix $A$, an eigenvector $\mathbf{v}$ is a non-zero vector that $A$ merely scales — it doesn't change direction:

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

The scalar $\lambda$ is the eigenvalue — the factor by which $A$ stretches (or compresses, or flips) the eigenvector.

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

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

This is the characteristic equation — a polynomial in $\lambda$ whose roots are the eigenvalues.

For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:

$$\det(A - \lambda I) = \lambda^2 - (a+d)\lambda + (ad - bc) = 0$$

The sum of eigenvalues equals the trace ($a + d$); the product equals the determinant ($ad - bc$). These shortcuts are useful for quick checks.

Economic applications:

• Input-output models (Leontief): the dominant eigenvalue of the technology matrix determines whether the economy can produce a net surplus — if it exceeds 1, some sector consumes more than it produces.
• Dynamic systems: eigenvalues of the transition matrix determine stability. If all $|\lambda| < 1$, the system converges to a steady state; if any $|\lambda| > 1$, it explodes.
• Markov chains: the stationary distribution is the eigenvector corresponding to $\lambda = 1$.
• Principal Component Analysis: eigenvectors of the covariance matrix are the principal components — the directions of greatest variation in the data.

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

The $2 \times 2$ shortcut

Tip: For any $2 \times 2$ matrix, skip the determinant expansion: write $\lambda^2 - (\text{trace})\lambda + \det = 0$ directly, then verify roots against trace and determinant. Twenty seconds, self-checking.