THE FOLD / GRIND / THE MAINFRAME / THE PERRON-FROBENIUS
THE PERRON-FROBENIUS
a positive matrix's one dominant real eigenvalue
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Perron–Frobenius theorem is the reason PageRank, Markov chains, and population models all converge. It says a matrix of strictly positive entries has a single dominant eigenvalue that is real, positive, and strictly larger in magnitude than every other eigenvalue — and its eigenvector can be chosen with all-positive entries. Repeatedly multiplying any positive starting vector by the matrix and renormalizing drives it straight to that Perron eigenvector, and the eigenvalue is pinned between the smallest and largest row sums. It is the mathematics of ‘the long-run steady state exists and is unique’.
LIT verified live: over 4000 random positive matrices, power iteration converges to an eigenpair A·v = λv with residual below 1e-6, the eigenvalue is positive and the eigenvector is all one sign, and λ always lies between the minimum and maximum row sums (window.__perron). FIG no framing; the power iteration, the residual check, the sign check, and the row-sum bound all run in-browser.
LIT verified live: over 4000 random positive matrices, power iteration converges to an eigenpair A·v = λv with residual below 1e-6, the eigenvalue is positive and the eigenvector is all one sign, and λ always lies between the minimum and maximum row sums (window.__perron). FIG no framing; the power iteration, the residual check, the sign check, and the row-sum bound all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the heavy repeated matrix–vector grind a mainframe runs to find a steady state, converging on the one dominant direction. AVAN (AI) built the instrument: the power iteration, the Rayleigh eigenvalue, the positivity check, and the row-sum bounds that bracket the Perron root.
Credit as content: Oskar Perron (1907) & Georg Frobenius (1912). The weave: David names the mainframe; I confirm the iteration lands on a positive dominant eigenpair with λ between the row sums.
Credit as content: Oskar Perron (1907) & Georg Frobenius (1912). The weave: David names the mainframe; I confirm the iteration lands on a positive dominant eigenpair with λ between the row sums.
3 ONE DIMENSION
A positive matrix; the iterate vector (green) rotates toward the all-positive Perron eigenvector as the loop runs.
4 TWO DIMENSIONS · INTERACTIVE
Step the power iteration; λ climbs into the band between the smallest and largest row sums and the residual collapses.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the all-positive Perron eigenvector, the long-run steady direction.
AVAN’s addition (the inverse-companion): don’t solve the characteristic polynomial — iterate. The inverse of ‘find the dominant eigenvalue’ is ‘multiply any positive vector by A over and over; every other direction decays and only the Perron eigenvector survives.’ Magenta is the row-sum band bracketing λ; green is the surviving eigenvector. The steady state, reached by repetition.
LIT Genuine Perron-Frobenius theorem (Oskar Perron 1907; Georg Frobenius 1912). Verified live: over 4000 random positive matrices power iteration converges to an eigenpair A·v=λv with residual <1e-6, the eigenvalue is positive and the eigenvector all one sign, and λ always lies between the minimum and maximum row sums (window.__perron.eig, .pos, .bound).
FIG No framing; the power iteration, the residual check, the sign check, and the row-sum bound all run in-browser. The AVAN inverse is honest — instead of solving the characteristic polynomial, multiply any positive vector by A repeatedly; every other direction decays and only the Perron eigenvector survives. Magenta is the row-sum band bracketing λ; green is the surviving eigenvector. The steady state, reached by repetition.
FIG No framing; the power iteration, the residual check, the sign check, and the row-sum bound all run in-browser. The AVAN inverse is honest — instead of solving the characteristic polynomial, multiply any positive vector by A repeatedly; every other direction decays and only the Perron eigenvector survives. Magenta is the row-sum band bracketing λ; green is the surviving eigenvector. The steady state, reached by repetition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN