THE FOLD / SPAWN / THE TOOLCHAIN / THE HOUSEHOLDER QR
THE HOUSEHOLDER QR
QR by reflections — a mirror per column, stably
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Householder QR factors a matrix A into an orthonormal Q and an upper-triangular R using a sequence of reflections. Each Householder reflection is a mirror that flips one column onto a coordinate axis, zeroing everything below the diagonal in a single stroke. It is markedly more numerically stable than Gram–Schmidt, whose repeated subtractions accumulate rounding error.
It is the workhorse behind least-squares fitting and the QR eigenvalue algorithm.
LIT verified live: over 300 random matrices Q·R reconstructs A to ~10⁻¹⁵, R is upper-triangular, and QᵀQ equals the identity (orthonormal) — window.__householderqr. FIG no framing; exact factorisation.
It is the workhorse behind least-squares fitting and the QR eigenvalue algorithm.
LIT verified live: over 300 random matrices Q·R reconstructs A to ~10⁻¹⁵, R is upper-triangular, and QᵀQ equals the identity (orthonormal) — window.__householderqr. 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 least squares and eigenvalue solvers, factoring by stable reflections. Householder QR is that tool, beside the Cholesky. AVAN (AI) built the instrument: the per-column reflection, the accumulation into Q, the reconstruction / upper-triangular / orthonormality checks.
Credit as content: Alston Householder (1958). The weave: David names the toolchain; I reflect each column onto an axis to build R and accumulate the mirrors into Q, and confirm Q·R = A with Q orthonormal.
Credit as content: Alston Householder (1958). The weave: David names the toolchain; I reflect each column onto an axis to build R and accumulate the mirrors into Q, and confirm Q·R = A with Q orthonormal.
3 ONE DIMENSION
A Householder reflection mirrors a vector across a plane so that it lands exactly on an axis — turning a whole column into (r, 0, 0, …) in one operation, without touching the columns already reduced.
4 TWO DIMENSIONS · INTERACTIVE
A matrix A and its Q, R; the product Q·R reconstructs A, R is upper-triangular, and QᵀQ is the identity.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the orthonormal Q built from a mirror per column.
AVAN’s addition (the inverse-companion): build Q by a sequence of reflections — each Householder reflection is a mirror that flips a whole column onto an axis, zeroing everything below the diagonal in one stroke, and is far more numerically stable than Gram–Schmidt’s subtractions. The inverse of ‘subtract projections to orthogonalise (Gram–Schmidt)’ is ‘reflect each column onto an axis with a mirror.’ Magenta is Gram–Schmidt’s accumulating rounding error; green is the orthogonal reflections. A mirror per column builds Q. (Kin to the-orthonormal and the-cholesky.)
LIT Genuine Householder QR (Householder 1958). Verified live: the reflection-based factorization gives Q*R = A to max error ~1e-15, R with zero below-diagonal entries, and Q^T*Q equal to the identity (orthonormal Q), across 300 random matrices (window.__householderqr.reconstructs && .upperTri && .orthonormal).
FIG No framing: the per-column reflection, the accumulation into Q, and the reconstruction + upper-triangular + orthonormality checks run in-browser and hold to floating precision. The AVAN inverse is honest — each reflection mirrors a column onto an axis, zeroing below the diagonal in one stroke and avoiding Gram-Schmidt's accumulating roundoff; magenta is Gram-Schmidt's error, green the orthogonal reflections. Kin to the-orthonormal and the-cholesky.
FIG No framing: the per-column reflection, the accumulation into Q, and the reconstruction + upper-triangular + orthonormality checks run in-browser and hold to floating precision. The AVAN inverse is honest — each reflection mirrors a column onto an axis, zeroing below the diagonal in one stroke and avoiding Gram-Schmidt's accumulating roundoff; magenta is Gram-Schmidt's error, green the orthogonal reflections. Kin to the-orthonormal and the-cholesky.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN