THE FOLD / LOOT / THE DROP / THE CAUCHY-BINET
THE CAUCHY-BINET
a product determinant equal to a sum of minor products
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Cauchy–Binet formula is the determinant identity for non-square matrices. If A is m×n and B is n×m with m ≤ n, the product AB is square, and det(AB) = ∑S det(A[:,S])·det(B[S,:]), where the sum runs over every choice of m columns S out of n. The determinant of a product decomposes into a sum over all m×m minors. Its most famous special case, with B = AT, gives det(AAT) = ∑S det(AS)² — the Gram determinant is a sum of squared minors, which is why it’s never negative and equals the squared volume of the row parallelepiped.
LIT verified live with exact integer arithmetic: for thousands of random integer matrices, det(AB) computed directly equals the sum ∑S det(A[:,S])·det(B[S,:]) over all column-subsets, and det(AAT) equals ∑S det(AS)² exactly (window.__cauchybinet). FIG no framing; the product determinant and the minor-sum are computed by different routes and agree exactly.
LIT verified live with exact integer arithmetic: for thousands of random integer matrices, det(AB) computed directly equals the sum ∑S det(A[:,S])·det(B[S,:]) over all column-subsets, and det(AAT) equals ∑S det(AS)² exactly (window.__cauchybinet). FIG no framing; the product determinant and the minor-sum are computed by different routes and agree exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-drop — the loot drop where every m-column subset drops in its own minor-product, and the whole pile sums to the single product determinant. AVAN (AI) built the instrument: the product determinant, the sum over all column-subset minors, and their exact agreement.
Credit as content: Augustin-Louis Cauchy and Jacques Binet (1812). The weave: David names the drop; I confirm det(AB) equals the sum of paired minors.
Credit as content: Augustin-Louis Cauchy and Jacques Binet (1812). The weave: David names the drop; I confirm det(AB) equals the sum of paired minors.
3 ONE DIMENSION
Matrix A (m×n); each choice of m columns gives a minor — the product determinant sums over all of them.
4 TWO DIMENSIONS · INTERACTIVE
New matrices; det(AB) is compared to the sum of paired minors Σ det(A[:,S])·det(B[S,:]).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: det(AB), one number.
AVAN’s addition (the inverse-companion): don’t multiply then take a determinant — sum over subsets. The inverse of ‘det(AB)’ is ‘the sum of paired m×m minors over every column-subset’, and with B = AT it becomes a sum of squares — the Gram determinant. Magenta are the subset minors; green is the product determinant they sum to. A determinant scattered across all its minors.
LIT Genuine Cauchy–Binet formula (Augustin-Louis Cauchy & Jacques Binet, 1812). Verified live with exact BigInt: for ~1200 random integer matrices, det(AB) equals Σ_S det(A[:,S])·det(B[S,:]) over all m-column subsets, and det(AAᵀ) equals Σ_S det(A_S)² (window.__cauchybinet.ok, .okSym, .cnt).
FIG No framing; the product determinant and the minor-sum are computed by different routes and agree exactly. The AVAN inverse is honest — instead of multiplying then taking a determinant, sum over subsets: the inverse of 'det(AB)' is 'the sum of paired m×m minors over every column-subset', and with B=Aᵀ it becomes a sum of squares (the Gram determinant). Magenta are the subset minors; green is the product determinant they sum to. A determinant scattered across all its minors.
FIG No framing; the product determinant and the minor-sum are computed by different routes and agree exactly. The AVAN inverse is honest — instead of multiplying then taking a determinant, sum over subsets: the inverse of 'det(AB)' is 'the sum of paired m×m minors over every column-subset', and with B=Aᵀ it becomes a sum of squares (the Gram determinant). Magenta are the subset minors; green is the product determinant they sum to. A determinant scattered across all its minors.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN