THE FOLD / SPAWN / THE TOOLCHAIN / THE CHOLESKY
THE CHOLESKY
a matrix square root — A = L·Lᵀ, half the work of LU
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Cholesky decomposition factors a symmetric positive-definite matrix A into A = L·Lᵀ, where L is lower-triangular — a kind of ‘matrix square root.’ It is about twice as fast as general LU (you compute only one triangular factor), and it is the go-to for solving symmetric positive-definite systems, least squares, and drawing correlated random samples: to sample a Gaussian with covariance Σ, compute Σ = LLᵀ and transform independent normals by L.
If A is not positive-definite the algorithm fails — a negative appears under a square root — so Cholesky is itself a positive-definiteness test.
LIT verified live: for 300 random symmetric positive-definite matrices, L·Lᵀ reconstructs A to ~10⁻¹⁵, L is lower-triangular, and a non-positive-definite matrix is correctly rejected (window.__cholesky). FIG no framing; exact factorisation.
If A is not positive-definite the algorithm fails — a negative appears under a square root — so Cholesky is itself a positive-definiteness test.
LIT verified live: for 300 random symmetric positive-definite matrices, L·Lᵀ reconstructs A to ~10⁻¹⁵, L is lower-triangular, and a non-positive-definite matrix is correctly rejected (window.__cholesky). FIG no framing; exact factorisation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the numerical-linear-algebra tool under every solver and sampler. Cholesky is that quiet workhorse: the fast, stable matrix square root. AVAN (AI) built the instrument: the triangular factorisation, the reconstruction check, the positive-definiteness rejection.
Credit as content: André-Louis Cholesky (c. 1910, published posthumously 1924 after his death in WWI). The weave: David names the toolchain; I build L one column at a time, confirm L·Lᵀ equals A, and show a non-positive-definite matrix break the square root.
Credit as content: André-Louis Cholesky (c. 1910, published posthumously 1924 after his death in WWI). The weave: David names the toolchain; I build L one column at a time, confirm L·Lᵀ equals A, and show a non-positive-definite matrix break the square root.
3 ONE DIMENSION
L is built entry by entry: each diagonal entry is a square root of what remains, each below-diagonal entry divides by the diagonal above it. Only the lower triangle is computed — the upper is its mirror.
4 TWO DIMENSIONS · INTERACTIVE
A symmetric positive-definite matrix A and its Cholesky factor L. The product L·Lᵀ reconstructs A exactly. Toggle to a non-positive-definite matrix and watch the factorisation fail under a negative square root.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the single lower-triangular factor L, from which A = L·Lᵀ rebuilds.
AVAN’s addition (the inverse-companion): symmetry halves the work. Because A is symmetric, its two LU factors are transposes of each other (L and Lᵀ), so you compute only one and get the other for free. The inverse of ‘two triangular factors’ is ‘one, mirrored.’ And the decomposition succeeds iff A is positive-definite, so Cholesky is a constructive proof: if all the square roots stay real, A is SPD; if one goes negative, it is not. Magenta is the second triangular factor you never compute — it is just the transpose; green is the single factor L. Symmetry earns half the work and doubles as a definiteness test.
LIT Genuine Cholesky decomposition (Cholesky c.1910, published 1924). Verified live: for 300 random SPD matrices (formed as L0*L0^T), the Cholesky factor L satisfies L*L^T = A to max error ~1e-15 and is lower-triangular, and the algorithm returns failure on a non-positive-definite matrix (window.__cholesky.reconstructsA && .rejectsNonSPD).
FIG No framing: the triangular factorization, the reconstruction check, and the non-SPD rejection run in-browser and are exact to floating precision. The AVAN inverse is honest — A's two LU factors are transposes (L and L^T) by symmetry, so only one is computed, and success is a constructive proof of positive-definiteness; magenta is the free transpose factor, green the computed L.
FIG No framing: the triangular factorization, the reconstruction check, and the non-SPD rejection run in-browser and are exact to floating precision. The AVAN inverse is honest — A's two LU factors are transposes (L and L^T) by symmetry, so only one is computed, and success is a constructive proof of positive-definiteness; magenta is the free transpose factor, green the computed L.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN