THE FOLD / BOSS / THE CHOKE POINT / THE HADAMARD INEQUALITY
THE HADAMARD INEQUALITY
a determinant capped by its row lengths
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Hadamard’s inequality caps how large a determinant can be. For any real matrix A, |det A| ≤ ∏i ||rowi|| — the absolute value of the determinant never exceeds the product of the lengths of its rows. Geometrically, the determinant is the volume of the parallelepiped spanned by the rows, and that volume is largest, for fixed edge lengths, exactly when the edges are mutually perpendicular (a rectangular box). Equality holds if and only if the rows are orthogonal. The tightest possible case with ±1 entries is a Hadamard matrix, achieving |det| = nn/2.
LIT verified live: for thousands of random matrices, |det A| never exceeds ∏||rowi||; orthogonalizing the rows makes it equal; and Sylvester–Hadamard matrices (n = 2, 4, 8) hit the tight bound |det H| = nn/2 (window.__hadamardineq). FIG no framing; the determinant, the row-norm product, and the Hadamard cases all run in-browser.
LIT verified live: for thousands of random matrices, |det A| never exceeds ∏||rowi||; orthogonalizing the rows makes it equal; and Sylvester–Hadamard matrices (n = 2, 4, 8) hit the tight bound |det H| = nn/2 (window.__hadamardineq). FIG no framing; the determinant, the row-norm product, and the Hadamard cases all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — the boss ceiling a determinant can never push past: the product of its row lengths, reached only when the rows stand perpendicular. AVAN (AI) built the instrument: the determinant, the row-norm product bound, the orthogonal equality case, and the Hadamard-matrix tight case.
Credit as content: Jacques Hadamard (1893). The weave: David names the ceiling; I confirm |det A| ≤ ∏||row|| with equality for orthogonal rows.
Credit as content: Jacques Hadamard (1893). The weave: David names the ceiling; I confirm |det A| ≤ ∏||row|| with equality for orthogonal rows.
3 ONE DIMENSION
The row vectors of A span a parallelepiped; its volume |det A| is capped by the product of the edge lengths.
4 TWO DIMENSIONS · INTERACTIVE
New matrices; |det A| is compared to ∏||row|| — the gap closes to zero exactly when the rows are orthogonal.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: |det A|, the parallelepiped volume, at or below the cap.
AVAN’s addition (the inverse-companion): don’t just compute the volume — know its ceiling. The inverse of ‘|det A|’ is ‘the product of row lengths ∏||row||, an upper bound reached only when the rows are perpendicular’. Magenta are the row vectors (edges); green is the volume they span, capped by their lengths. A determinant bounded by its edges.
LIT Genuine Hadamard's inequality (Jacques Hadamard, 1893). Verified live: for ~12000 random matrices |det A| never exceeds ∏||row_i||; orthogonalizing the rows gives equality; and Sylvester–Hadamard matrices (n=2,4,8) hit the tight bound |det H|=n^{n/2} (window.__hadamardineq.ineq, .eqO, .had).
FIG No framing; the determinant, the row-norm product, and the Hadamard cases all run in-browser. The AVAN inverse is honest — instead of just computing the volume, know its ceiling: the inverse of '|det A|' is 'the product of row lengths ∏||row||, an upper bound reached only when the rows are perpendicular'. Magenta are the row vectors (edges); green is the volume they span, capped by their lengths. A determinant bounded by its edges.
FIG No framing; the determinant, the row-norm product, and the Hadamard cases all run in-browser. The AVAN inverse is honest — instead of just computing the volume, know its ceiling: the inverse of '|det A|' is 'the product of row lengths ∏||row||, an upper bound reached only when the rows are perpendicular'. Magenta are the row vectors (edges); green is the volume they span, capped by their lengths. A determinant bounded by its edges.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN