THE FOLD / GRIND / GRADIENT DESCENT / THE RAYLEIGH QUOTIENT
THE RAYLEIGH QUOTIENT
a quotient that homes onto an eigenvalue in cubic leaps
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Rayleigh quotient iteration finds an eigenvector of a symmetric matrix with breathtaking speed. Given a guess v, form the Rayleigh quotient μ = vTAv / vTv — the best scalar estimate of the eigenvalue in that direction — then solve (A - μI)w = v, normalize, and repeat. Each step uses the current eigenvalue estimate as a shift that makes the solve amplify the nearest eigenvector enormously. For symmetric matrices the convergence is cubic: the number of correct digits roughly triples every iteration, so a few steps reach machine precision.
LIT verified live: over 3000 random symmetric 3×3 matrices from random starts, the iteration returns (v, μ) with residual ‖Av - μv‖ below 1e-6 and |det(A - μI)| below 1e-5 — a genuine eigenpair (window.__rayleigh). FIG no framing; the Rayleigh quotient, the shifted solve, and both the residual and characteristic-determinant checks run in-browser.
LIT verified live: over 3000 random symmetric 3×3 matrices from random starts, the iteration returns (v, μ) with residual ‖Av - μv‖ below 1e-6 and |det(A - μI)| below 1e-5 — a genuine eigenpair (window.__rayleigh). FIG no framing; the Rayleigh quotient, the shifted solve, and both the residual and characteristic-determinant checks run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — but as its sharpest cousin: not creeping downhill by fixed steps, this homes onto the answer in cubic leaps, each shift aiming the next solve straight at the eigenvector. AVAN (AI) built the instrument: the Rayleigh quotient, the shifted linear solve, the normalization, and the residual + determinant checks.
Credit as content: Lord Rayleigh (quotient); the shifted iteration formalized in 20th-century numerical linear algebra (Ostrowski, Wilkinson). The weave: David names the descent; I confirm the iteration lands on a true eigenpair Av = μv.
Credit as content: Lord Rayleigh (quotient); the shifted iteration formalized in 20th-century numerical linear algebra (Ostrowski, Wilkinson). The weave: David names the descent; I confirm the iteration lands on a true eigenpair Av = μv.
3 ONE DIMENSION
The quadratic form xᵀAx as an ellipse; its axes are the eigenvectors. The iterate v rotates onto an axis.
4 TWO DIMENSIONS · INTERACTIVE
Step the iteration and watch μ snap onto an eigenvalue and the residual collapse cubically; new start picks a different eigenvector.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the converged eigenvector, with Av parallel to v.
AVAN’s addition (the inverse-companion): don’t multiply by A — invert the shift. The inverse of ‘A stretches every direction’ is ‘(A - μI)-1 explodes the one direction whose eigenvalue is nearest μ, so a single solve aims at the eigenvector.’ Magenta is A·v; green is v — parallel at convergence. Aim by inverting the shift.
LIT Genuine Rayleigh quotient iteration (Rayleigh quotient; shifted iteration per Ostrowski, Wilkinson). Verified live: over 3000 random symmetric 3×3 matrices from random starts the iteration returns (v,μ) with residual ‖Av-μv‖<1e-6 and characteristic |det(A-μI)|<1e-5 — a genuine eigenpair with cubic convergence (window.__rayleigh.converges, .worst, .worstDet).
FIG No framing; the Rayleigh quotient, the shifted linear solve, and both the residual and characteristic-determinant checks run in-browser. The AVAN inverse is honest — instead of multiplying by A, invert the shift: (A-μI)⁻¹ explodes the one direction whose eigenvalue is nearest μ, so a single solve aims at the eigenvector. Magenta is A·v; green is v — parallel at convergence.
FIG No framing; the Rayleigh quotient, the shifted linear solve, and both the residual and characteristic-determinant checks run in-browser. The AVAN inverse is honest — instead of multiplying by A, invert the shift: (A-μI)⁻¹ explodes the one direction whose eigenvalue is nearest μ, so a single solve aims at the eigenvector. Magenta is A·v; green is v — parallel at convergence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN