THE FOLD / LOOT / THE MINT / THE VANDERMONDE DETERMINANT
THE VANDERMONDE DETERMINANT
a determinant that factors into pairwise differences
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Vandermonde determinant gives a stunningly clean answer to a messy-looking question. Build a matrix whose rows are the powers of some numbers x0, x1, …, xn-1 — row i is (1, xi, xi², …, xin-1). Its determinant, which looks like it should be a horrible polynomial, factors perfectly into a product of all pairwise differences: det = ∏i<j (xj - xi). So the determinant is zero exactly when two of the numbers coincide — which is why polynomial interpolation through distinct points always has a unique solution. It underlies interpolation, coding theory (Reed–Solomon), and the theory of symmetric functions.
LIT verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏i<j(xj - xi) to ~1e-9 (window.__vandermonde). FIG no framing; the determinant and the product of differences are computed independently in-browser.
LIT verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏i<j(xj - xi) to ~1e-9 (window.__vandermonde). FIG no framing; the determinant and the product of differences are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — the loot: a whole determinant minted, cleanly, out of nothing but the pairwise gaps between the nodes. AVAN (AI) built the instrument: the Vandermonde matrix, its determinant, and the product-of-differences formula.
Credit as content: Alexandre-Théophile Vandermonde. The weave: David names the mint; I confirm det = ∏i<j(xj - xi).
Credit as content: Alexandre-Théophile Vandermonde. The weave: David names the mint; I confirm det = ∏i<j(xj - xi).
3 ONE DIMENSION
A Vandermonde matrix of powers; its determinant equals the product of all pairwise node differences.
4 TWO DIMENSIONS · INTERACTIVE
Cycle node sets; the matrix determinant is checked against the product of pairwise differences.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the determinant, equal to the product of pairwise gaps.
AVAN’s addition (the inverse-companion): don’t expand the determinant — read it off the gaps. The inverse of ‘det of the power matrix’ is ‘the product ∏i<j(xj - xi) of pairwise differences’ — zero exactly when two nodes collide. Magenta are the pairwise node differences; green is the determinant they multiply to. A determinant that is just the gaps.
LIT Genuine Vandermonde determinant (Alexandre-Théophile Vandermonde). Verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏_{i
FIG No framing; the determinant and the product of differences are computed independently in-browser. The AVAN inverse is honest — instead of expanding the determinant, read it off the gaps: the inverse of 'det of the power matrix' is 'the product ∏_{i
FIG No framing; the determinant and the product of differences are computed independently in-browser. The AVAN inverse is honest — instead of expanding the determinant, read it off the gaps: the inverse of 'det of the power matrix' is 'the product ∏_{i
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN