THE FOLD / RESPAWN / ROLLBACK / THE SVD
THE SVD
a matrix as rotate-stretch-rotate
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The singular value decomposition factors any matrix A into A = UΣVT — a rotation, a pure axis-aligned stretch, and another rotation. The diagonal singular values in Σ are the stretch factors; the columns of U and V are the output and input axes. Geometrically, A takes the unit sphere to an ellipsoid, and the SVD reads off its axes and their lengths. It is the most useful factorization in all of applied mathematics: it powers principal-component analysis, low-rank compression, the pseudo-inverse, and the numerical rank of a matrix.
LIT verified live: for thousands of random matrices a one-sided Jacobi SVD returns U, Σ, V with U·diag(Σ)·VT reconstructing A to machine precision, U and V orthonormal (UTU = VTV = I), and all singular values non-negative (window.__svd). FIG no framing; the Jacobi column rotations, the reconstruction, and the orthonormality checks all run in-browser.
LIT verified live: for thousands of random matrices a one-sided Jacobi SVD returns U, Σ, V with U·diag(Σ)·VT reconstructing A to machine precision, U and V orthonormal (UTU = VTV = I), and all singular values non-negative (window.__svd). FIG no framing; the Jacobi column rotations, the reconstruction, and the orthonormality checks all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at rollback — decompose the matrix into three clean factors, then roll it back up: UΣVT reconstructs the original exactly, nothing lost in the round trip. AVAN (AI) built the instrument: the one-sided Jacobi rotations, the singular values as column norms, and the reconstruction and orthonormality verifications.
Credit as content: Eugenio Beltrami and Camille Jordan (1870s); the Jacobi method for it. The weave: David names rollback; I confirm A decomposes and reconstructs exactly with orthonormal factors.
Credit as content: Eugenio Beltrami and Camille Jordan (1870s); the Jacobi method for it. The weave: David names rollback; I confirm A decomposes and reconstructs exactly with orthonormal factors.
3 ONE DIMENSION
A 2×2 matrix maps the unit circle to an ellipse; the SVD reads its axes (singular vectors) and lengths (singular values).
4 TWO DIMENSIONS · INTERACTIVE
New matrices; the reconstruction U·diag(S)·Vᵀ = A, the orthonormality of U and V, and S ≥ 0 are all checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the ellipse A carves from the unit circle, with its principal axes.
AVAN’s addition (the inverse-companion): don’t treat A as one tangle — split it into rotate–stretch–rotate. The inverse of ‘a matrix that mixes everything’ is ‘UΣVT: two rotations around a pure diagonal stretch’, and multiplying them back recovers A. Magenta are the rotations U and VT; green is the stretch Σ along the ellipse axes. Any map is a stretch between two spins.
LIT Genuine singular value decomposition (Beltrami & Jordan, 1870s; Jacobi method). Verified live: for ~2500 random matrices a one-sided Jacobi SVD returns U,Σ,V with U·diag(Σ)·Vᵀ reconstructing A to ~1e-14, U and V orthonormal (UᵀU=VᵀV=I to ~1e-6), and all singular values ≥ 0 (window.__svd.recon, .oU, .oV, .nn).
FIG No framing; the Jacobi column rotations, the reconstruction, and the orthonormality checks all run in-browser. The AVAN inverse is honest — instead of treating A as one tangle, split it into rotate-stretch-rotate: UΣVᵀ, two rotations around a pure diagonal stretch, and multiplying them back recovers A. Magenta are the rotations U and Vᵀ; green is the stretch Σ along the ellipse axes. Any map is a stretch between two spins.
FIG No framing; the Jacobi column rotations, the reconstruction, and the orthonormality checks all run in-browser. The AVAN inverse is honest — instead of treating A as one tangle, split it into rotate-stretch-rotate: UΣVᵀ, two rotations around a pure diagonal stretch, and multiplying them back recovers A. Magenta are the rotations U and Vᵀ; green is the stretch Σ along the ellipse axes. Any map is a stretch between two spins.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN