Power Iteration: Watching a Vector Converge to a Dominant Eigenvector (2D)

Matrix A

a (row 1, col 1) 0.90

b (row 1, col 2) 0.60

c (row 2, col 1) 0.30

d (row 2, col 2) 1.40

Iteration

Steps per second 2.0

Starting angle θ₀ 35°

Live Readout

Iteration n

Vector angle θ

0.0°

Rayleigh est. λ

0.000

Error vs analytic

0.000

Eigenvalues (analytic)

–

det(A) / tr(A)

–

How it works

The power method finds the dominant eigenvector of A without solving the characteristic polynomial: start from any unit vector v₀, repeatedly apply v_n+1 = A·v_n / |A·v_n|. If A has one eigenvalue strictly larger in magnitude than the rest, v_n converges to that eigenvector's direction, because each multiplication amplifies the dominant component exponentially faster than the others.

The eigenvalue is read off with the Rayleigh quotient λ ≈ v_nᵀ·A·v_n (v_n being unit length), which converges quadratically once v_n is close to the true eigenvector. The error panel plots |λ_est − λ_analytic| on a log scale, so the (roughly straight, then plummeting) convergence line is visible directly.

When the discriminant tr(A)² − 4·det(A) is negative, A has no real eigenvector at all — it acts as a rotation-and-scale. Power iteration then never settles: v_n spins around the origin forever at a fixed angular step, and the angle panel shows a rising ramp instead of a plateau.

Vector pane — the current unit vector v_n (cyan arrow) against the analytic eigen-directions (dashed, when real) and the unit circle A maps to (faint ellipse).
Angle pane — θ(v_n) per iteration; a flat line means convergence, a steady ramp means rotation (complex eigenvalues).
Convergence pane — |λ_est − λ_analytic| on a log axis; a falling line confirms the power method is honestly converging, not just visually settling.

Vector v_n Rayleigh estimate λ |error| (log)