Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Uniqueness — the ladder

One essay so far against this idea. A ladder is the distinct arguments that stand against one idea, and this one has room to grow.
  1. 6 × 6 × 6, rank 3 · 9 ≥ 81.00002 × 2 × 2, rank 3 · 6 < 80.02076 × 6 matrix, rank 3 · no condition0.1089worst agreement between two runs about the factors, 0 to 1every run fits to 3·10⁻¹³every run fits to 1.2·10⁻¹¹every run factorises to 2.4·10⁻¹⁵ and none agrees with anotherthe only thing that improvestensor, Kruskal holds1tensor, Kruskal fails0.021matrix0.11worst residual1.2·10⁻¹¹three successful fitsone recoverable answer

    A factorisation that is unique for once

    A rank-r factorisation of a matrix is never unique — AB is (AM)(M⁻¹B) for any invertible M, so no factor means anything on its own. For three indices a checkable condition on the factors' k-ranks makes the decomposition unique up to permuting and scaling the terms, and it holds generically.

    rung 1 · tensor

All ladders