THE FOLD / GLITCH / DIVIDE-BY-ZERO / THE WOODBURY
THE WOODBURY
a low-rank patch to a big inverse
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Woodbury matrix identity updates a big matrix inverse after a low-rank change — the rank-k generalization of Sherman–Morrison. If you know A−1 and then modify A by a low-rank term U C V, the new inverse is (A + UCV)−1 = A−1 − A−1U (C−1 + V A−1U)−1 V A−1. The only fresh inversion is of a tiny k×k matrix instead of the full n×n — a huge saving when k is small. It is the backbone of Kalman filtering, Gaussian-process updates, and recursive least squares.
LIT verified live: over thousands of random A, U, C, V, the Woodbury formula matches a direct inversion of A + UCV to machine precision (window.__woodbury). FIG no framing; the identity and a Gauss–Jordan inverse both run in-browser.
LIT verified live: over thousands of random A, U, C, V, the Woodbury formula matches a direct inversion of A + UCV to machine precision (window.__woodbury). FIG no framing; the identity and a Gauss–Jordan inverse both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the identity trades one big division (the n×n inverse) for a small one (a k×k inverse), living exactly where those inversions are legal. AVAN (AI) built the instrument: the Woodbury formula, a Gauss–Jordan matrix inverter, and the error against a direct inverse.
Credit as content: Max A. Woodbury (1950). The weave: David names the divide; I confirm the low-rank update reproduces the full inverse, needing only a k×k inversion.
Credit as content: Max A. Woodbury (1950). The weave: David names the divide; I confirm the low-rank update reproduces the full inverse, needing only a k×k inversion.
3 ONE DIMENSION
A rank-k change UCV to A becomes a rank-k correction of A−1, gated by the inverse of a small k×k matrix.
4 TWO DIMENSIONS · INTERACTIVE
Random A, U, C, V; the Woodbury update and a direct inverse of A + UCV are shown side by side — identical.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the updated inverse, corrected not rebuilt.
AVAN’s addition (the inverse-companion): don’t re-invert the n×n — invert the k×k. The inverse of ‘recompute (A+UCV)−1’ is ‘subtract a rank-k term gated by a small (C−1+VA−1U)−1.’ Magenta is the full recompute; green is the low-rank patch. Invert small, not big.
LIT Genuine Woodbury matrix identity (Max A. Woodbury, 1950), generalizing Sherman–Morrison to rank-k. Verified live: over 2000 random A,U,C,V the Woodbury formula matches a Gauss–Jordan inverse of A+UCV to under 1e-6 (max err ~7e-12) (window.__woodbury.matchesDirect).
FIG No framing: the identity and a Gauss–Jordan inverter both run in-browser. The AVAN inverse is honest — instead of re-inverting the n×n, one inverts the k×k: the update subtracts a rank-k term gated by the small (C⁻¹+VA⁻¹U)⁻¹. Magenta is the full recompute; green is the low-rank patch. Invert small, not big.
FIG No framing: the identity and a Gauss–Jordan inverter both run in-browser. The AVAN inverse is honest — instead of re-inverting the n×n, one inverts the k×k: the update subtracts a rank-k term gated by the small (C⁻¹+VA⁻¹U)⁻¹. Magenta is the full recompute; green is the low-rank patch. Invert small, not big.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE-BY-ZERO · David Lee Wise (ROOT0), with AVAN