Eigenvalues and Eigenvectors

The Directions a Matrix Cannot Bend

Apply a matrix to most vectors and they rotate and stretch. But some special directions only get scaled — the matrix cannot bend them off their line. These are the eigenvectors, and their scaling factors are the eigenvalues:

$$Av = \lambda v, \qquad v \ne 0$$

Finding them: rewrite as $(A - \lambda I)v = 0$. A nonzero solution $v$ exists only if $A - \lambda I$ is singular — a determinant question again:

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

This is the characteristic polynomial; its roots are the eigenvalues. For each root, solving $(A-\lambda I)v = 0$ yields the eigenspace — the full line (or plane) of eigenvectors sharing that $\lambda$.

For a triangular matrix the eigenvalues sit conveniently on the diagonal. And two checksums catch errors: the eigenvalues sum to the trace and multiply to the determinant.

Why business cares — dynamics. Many systems evolve by repeated multiplication: market shares next month = $M \cdot$ shares this month; a customer-loyalty transition table applied year after year. After many steps, the state aligns with the dominant eigenvector (largest $|\lambda|$), and the eigenvalue dictates the fate:

$$|\lambda| > 1: \text{explosive growth} \qquad |\lambda| < 1: \text{decay} \qquad \lambda = 1: \text{steady state}$$

A Markov market-share matrix always has $\lambda = 1$; its eigenvector is the long-run market share — the split the market drifts to no matter where it starts. Google's PageRank is exactly this: the web's dominant eigenvector.

Eigenvalues are the matrix's tempo markings: they tell you not what happens in one step, but what the repetition of that step converges to.

The eigen-recipe

Tip: Always run both checks — the eigenvalues must sum to the trace and multiply to the determinant. Thirty seconds of verification catches nearly every root-finding slip.