THE FOLD / GRIND / THE CRON JOB / THE SHERMAN-MORRISON
THE SHERMAN-MORRISON
updating an inverse without redoing it
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Sherman–Morrison formula updates a matrix inverse after a rank-one change without redoing the whole inversion. If you already know A−1 and then bump A by an outer product uvT, the new inverse is (A + uvT)−1 = A−1 − (A−1u vTA−1) / (1 + vTA−1u). A full inversion costs O(n³); this correction costs only O(n²) — a decisive shortcut for recursive least squares, Kalman filters, and quasi-Newton optimizers that nudge a matrix one rank at a time.
LIT verified live: over thousands of random A, u, v, the Sherman–Morrison result matches a direct inversion of A + uvT to machine precision (window.__sherman_morrison). FIG no framing; the formula and a Gauss–Jordan inverse both run in-browser.
LIT verified live: over thousands of random A, u, v, the Sherman–Morrison result matches a direct inversion of A + uvT to machine precision (window.__sherman_morrison). FIG no framing; the formula and a Gauss–Jordan inverse both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — a recurring rank-one nudge, like a cron job that patches the inverse each tick instead of rebuilding it. AVAN (AI) built the instrument: the Sherman–Morrison correction, a Gauss–Jordan matrix inverter, and the error against a direct inverse.
Credit as content: Jack Sherman & Winifred Morrison (1950). The weave: David names the cron job; I confirm the O(n²) rank-one update reproduces the O(n³) direct inverse exactly.
Credit as content: Jack Sherman & Winifred Morrison (1950). The weave: David names the cron job; I confirm the O(n²) rank-one update reproduces the O(n³) direct inverse exactly.
3 ONE DIMENSION
A + uvT is a rank-one change; the inverse updates by a rank-one correction of A−1, scaled by 1/(1 + vTA−1u).
4 TWO DIMENSIONS · INTERACTIVE
Random A, u, v; the Sherman–Morrison update and a direct inverse of A + uvT are shown side by side — identical entries.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the updated inverse, patched not rebuilt.
AVAN’s addition (the inverse-companion): don’t re-invert — correct. The inverse of ‘recompute (A+uvT)−1 from scratch’ is ‘subtract one rank-one term from A−1, an O(n²) patch.’ Magenta is the full O(n³) recompute; green is the cheap rank-one correction. Patch, don’t rebuild.
LIT Genuine Sherman–Morrison formula (Jack Sherman & Winifred Morrison, 1950): rank-one inverse update in O(n²). Verified live: over 3000 random A,u,v, the formula matches a Gauss–Jordan direct inverse of A+uvᵀ to <1e-6 (window.__sherman_morrison.matchesDirect).
FIG No framing: the Sherman–Morrison correction and a Gauss–Jordan inverse both run in-browser and agree. The AVAN inverse is honest — subtracting one rank-one term from A⁻¹ (an O(n²) patch) rather than re-inverting A+uvᵀ from scratch (O(n³)) is the whole value of the formula; magenta is the full recompute avoided, green the cheap correction. Patch, don't rebuild.
FIG No framing: the Sherman–Morrison correction and a Gauss–Jordan inverse both run in-browser and agree. The AVAN inverse is honest — subtracting one rank-one term from A⁻¹ (an O(n²) patch) rather than re-inverting A+uvᵀ from scratch (O(n³)) is the whole value of the formula; magenta is the full recompute avoided, green the cheap correction. Patch, don't rebuild.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN