THE FOLD / RESPAWN / THE RESURRECT / THE SYLVESTER INERTIA
THE SYLVESTER INERTIA
a signature invariant under congruence
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sylvester’s law of inertia says a symmetric matrix has an unchangeable ‘signature’. Any real symmetric matrix M can be transformed by congruence — M → PTMP for an invertible P — into many different-looking matrices. But the counts of positive, negative, and zero eigenvalues (the signature n+, n-, n0) never change. You can rescale and mix the coordinates however you like; the number of ‘plus’ and ‘minus’ directions of the quadratic form is a fixed invariant. It is what lets us classify quadratic forms and read the character (definite, indefinite) of a form from any convenient basis.
LIT verified live: for thousands of random symmetric matrices, the signature computed from the eigenvalue signs is unchanged after a random congruence PTMP; and the number of negative eigenvalues equals the number of sign changes in the sequence of leading principal minors (Jacobi’s criterion) — two independent computations of the same signature (window.__sylvesterinertia). FIG no framing; the eigenvalue signature, the congruence, and the minor-sign-change count all run in-browser.
LIT verified live: for thousands of random symmetric matrices, the signature computed from the eigenvalue signs is unchanged after a random congruence PTMP; and the number of negative eigenvalues equals the number of sign changes in the sequence of leading principal minors (Jacobi’s criterion) — two independent computations of the same signature (window.__sylvesterinertia). FIG no framing; the eigenvalue signature, the congruence, and the minor-sign-change count all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-resurrect — the invariant that resurrects unchanged after any congruence: mangle the matrix, and its signature comes back exactly as it was. AVAN (AI) built the instrument: the eigenvalue signature, the random congruence, and the leading-minor sign-change cross-check.
Credit as content: James Joseph Sylvester (1852); Carl Gustav Jacob Jacobi (minor criterion). The weave: David names the invariant; I confirm the signature survives congruence and matches the minor sign-changes.
Credit as content: James Joseph Sylvester (1852); Carl Gustav Jacob Jacobi (minor criterion). The weave: David names the invariant; I confirm the signature survives congruence and matches the minor sign-changes.
3 ONE DIMENSION
A symmetric matrix and its eigenvalue signs — the signature (n₊, n₋, n₀) that congruence can never change.
4 TWO DIMENSIONS · INTERACTIVE
New matrices; the signature is shown unchanged after a random congruence PᵀMP, and matched to the minor sign-changes.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the signature, invariant under every congruence.
AVAN’s addition (the inverse-companion): don’t read the matrix entries — count the signs. The inverse of ‘which symmetric matrix?’ is ‘its signature (n+, n-, n0)’, the one thing congruence cannot touch — also read off the sign changes in the leading minors. Magenta are the congruence-transformed matrices; green is the signature they all share. The invariant that survives every basis change.
LIT Genuine Sylvester's law of inertia (James Joseph Sylvester, 1852; Jacobi minor criterion). Verified live: for ~1200 random symmetric matrices, the signature (n₊,n₋,n₀) from eigenvalue signs is unchanged after a random congruence PᵀMP, and n₋ equals the number of sign changes in the leading principal minors (Jacobi) — two independent computations (window.__sylvesterinertia.inv, .jac).
FIG No framing; the eigenvalue signature, the congruence, and the minor-sign-change count all run in-browser. The AVAN inverse is honest — instead of reading the matrix entries, count the signs: the inverse of 'which symmetric matrix?' is 'its signature (n₊,n₋,n₀)', the one thing congruence cannot touch — also read off the sign changes in the leading minors. Magenta are the congruence-transformed matrices; green is the signature they all share. The invariant that survives every basis change.
FIG No framing; the eigenvalue signature, the congruence, and the minor-sign-change count all run in-browser. The AVAN inverse is honest — instead of reading the matrix entries, count the signs: the inverse of 'which symmetric matrix?' is 'its signature (n₊,n₋,n₀)', the one thing congruence cannot touch — also read off the sign changes in the leading minors. Magenta are the congruence-transformed matrices; green is the signature they all share. The invariant that survives every basis change.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN