THE FOLD / GRIND / THE MAINFRAME / THE GERSHGORIN
THE GERSHGORIN
trap every eigenvalue in a disc, without solving
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gershgorin’s circle theorem pins down where a matrix’s eigenvalues can be without computing them: every eigenvalue lies within at least one Gershgorin disc — a disc centered at a diagonal entry aii, with radius equal to the sum of the absolute values of the off-diagonal entries in that row. A few cheap sums bound the whole spectrum, which is invaluable for stability analysis and preconditioning.
LIT verified live: over 300 random symmetric matrices, every (Jacobi-computed) eigenvalue falls inside a Gershgorin disc, and the eigenvalues sum to the trace (window.__gershgorin). FIG no framing; exact containment.
LIT verified live: over 300 random symmetric matrices, every (Jacobi-computed) eigenvalue falls inside a Gershgorin disc, and the eigenvalues sum to the trace (window.__gershgorin). FIG no framing; exact containment.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the matrix engine that bounds a spectrum with a glance at the rows. Gershgorin’s theorem is that glance. AVAN (AI) built the instrument: the row-radius discs, an independent Jacobi eigensolver, the every-eigenvalue-in-a-disc check, and the sum-equals-trace sanity check.
Credit as content: Semyon Gershgorin (1931). The weave: David names the mainframe; I draw each row’s disc from its diagonal and off-diagonal sum, compute the eigenvalues independently, and confirm each one is trapped in a disc.
Credit as content: Semyon Gershgorin (1931). The weave: David names the mainframe; I draw each row’s disc from its diagonal and off-diagonal sum, compute the eigenvalues independently, and confirm each one is trapped in a disc.
3 ONE DIMENSION
Row i gives a disc: center at aii, radius = Σj≠i |aij|. Every eigenvalue of the matrix must sit inside the union of these discs — no eigenvalue escapes them all.
4 TWO DIMENSIONS · INTERACTIVE
A symmetric matrix, its Gershgorin discs on the real line, and its eigenvalues — each checked to lie inside a disc.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: eigenvalues trapped inside row-discs.
AVAN’s addition (the inverse-companion): bound the spectrum without solving for it — each row draws a disc (diagonal ± off-diagonal sum), and every eigenvalue must lie in the union. The inverse of ‘compute the eigenvalues to know their range’ is ‘read a disc off each row — they trap the whole spectrum.’ Magenta is the full eigen-solve you can skip for a bound; green is the row-disc that traps them. Localizing eigenvalues by rows.
LIT Genuine Gershgorin circle theorem (Gershgorin 1931). Verified live: over 300 random symmetric matrices, every eigenvalue (computed independently by a Jacobi eigensolver) lies within a Gershgorin disc (center a_ii, radius sum of off-diagonal magnitudes), and the eigenvalues sum to the trace as a sanity check (window.__gershgorin.allInDisc && .sumEqualsTrace).
FIG No framing: the row-radius discs, an independent Jacobi eigensolver, the every-eigenvalue-in-a-disc check, and the sum-equals-trace check run in-browser and hold. The AVAN inverse is honest — each row draws a disc (diagonal +/- off-diagonal sum) and every eigenvalue must lie in the union, bounding the spectrum with no eigen-solve; magenta is the full solve you can skip for a bound, green the row-disc that traps them. Localizing eigenvalues by rows.
FIG No framing: the row-radius discs, an independent Jacobi eigensolver, the every-eigenvalue-in-a-disc check, and the sum-equals-trace check run in-browser and hold. The AVAN inverse is honest — each row draws a disc (diagonal +/- off-diagonal sum) and every eigenvalue must lie in the union, bounding the spectrum with no eigen-solve; magenta is the full solve you can skip for a bound, green the row-disc that traps them. Localizing eigenvalues by rows.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN