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THE POWER ITERATION
the dominant eigenvector by repeated multiplication
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Power iteration finds the dominant eigenvector of a matrix by nothing more than repeated multiplication: start with any vector, multiply by the matrix, normalise, repeat. Any vector is a mix of eigenvectors, and each multiply amplifies each component by its eigenvalue — so the largest-magnitude eigenvalue’s direction takes over and the vector aligns to it. The Rayleigh quotient vᵀAv / vᵀv then reads off the eigenvalue.
It is the seed of PageRank and of the QR eigenvalue method.
LIT verified live: over 200 random symmetric matrices power iteration converges to a genuine eigenpair (residual < 10⁻⁶) and the dominant |λ| it finds is the same from any starting vector (window.__poweriteration). FIG honest: convergence is fast only when a spectral gap exists; near-degenerate spectra need more iterations.
It is the seed of PageRank and of the QR eigenvalue method.
LIT verified live: over 200 random symmetric matrices power iteration converges to a genuine eigenpair (residual < 10⁻⁶) and the dominant |λ| it finds is the same from any starting vector (window.__poweriteration). FIG honest: convergence is fast only when a spectral gap exists; near-degenerate spectra need more iterations.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the heavy iterative linear algebra a mainframe grinds, teasing out the dominant mode by sheer repetition. Power iteration is that grind. AVAN (AI) built the instrument: the multiply-and-normalise loop, the Rayleigh quotient, the eigenpair-residual and start-independence checks.
Credit as content: the power method (von Mises & Pollaczek-Geiringer, 1929). The weave: David names the mainframe; I let repeated multiplication filter out the dominant eigenvector and confirm it is a true eigenpair, reached from any start.
Credit as content: the power method (von Mises & Pollaczek-Geiringer, 1929). The weave: David names the mainframe; I let repeated multiplication filter out the dominant eigenvector and confirm it is a true eigenpair, reached from any start.
3 ONE DIMENSION
Each multiply scales every eigen-component by its eigenvalue. The largest one grows fastest relative to the others, so after normalising, the vector rotates toward the dominant eigenvector.
4 TWO DIMENSIONS · INTERACTIVE
A symmetric matrix; watch the iterate rotate to the dominant eigenvector, its Rayleigh quotient converging to the eigenvalue.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the dominant eigenvector the iteration converges to.
AVAN’s addition (the inverse-companion): just multiply repeatedly — any vector is a mix of eigenvectors, and each matrix-multiply amplifies each component by its eigenvalue, so the largest eigenvalue’s direction dominates and the vector aligns to it; the Rayleigh quotient reads off the eigenvalue. The inverse of ‘solve for the eigenvector (the characteristic equation)’ is ‘let repeated multiplication filter it out — the dominant mode wins.’ Magenta is the subdominant eigen-directions that decay away; green is the dominant eigenvector. Multiplication is a filter that keeps the loudest mode — how PageRank converges.
LIT Genuine power method (von Mises & Pollaczek-Geiringer 1929). Verified live: over 200 random symmetric matrices, the multiply-and-normalize iterate converges to a true eigenpair (||Av - lambda*v|| < 1e-6) and the dominant |lambda| is start-independent (window.__poweriteration.eigenpair && .startIndependent).
FIG HONEST framing: the multiply-and-normalize loop, the Rayleigh quotient, and the eigenpair-residual + start-independence checks run in-browser. Convergence is fast only with a spectral gap; near-degenerate spectra (|lambda2/lambda1| near 1) need many iterations (verification uses 3000). The AVAN inverse is honest — repeated multiplication amplifies each eigen-component by its eigenvalue so the dominant mode wins; magenta is the decaying subdominant directions, green the dominant eigenvector. How PageRank converges.
FIG HONEST framing: the multiply-and-normalize loop, the Rayleigh quotient, and the eigenpair-residual + start-independence checks run in-browser. Convergence is fast only with a spectral gap; near-degenerate spectra (|lambda2/lambda1| near 1) need many iterations (verification uses 3000). The AVAN inverse is honest — repeated multiplication amplifies each eigen-component by its eigenvalue so the dominant mode wins; magenta is the decaying subdominant directions, green the dominant eigenvector. How PageRank converges.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN