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LIT-330: The approximation of one matrix by another of lower rank

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StatusActive — read in full 2026-09-29 (NOTE-301); worth reading as the short, readable origin of the Frobenius-norm low-rank approximation theorem in the form still used. The proof is a first-order stationarity argument plus an unproved simultaneous-diagonalisation theorem, and it covers only the least-squares norm. For the general unitarily-invariant-norm statement, cite Mirsky (1960, not in the record).
Version2
Tagsmathematics, representation-learning
Cited as “paper” byNOTE-301 — The approximation of one matrix by another of lower rank
Cited as “source” byTHEORY-503 — The nearest rank-r matrix to a real matrix in the least-squares norm keeps its r largest singular values and zeroes the rest; it is unique when the r-th and (r+1)-th singular values differ, its squared error is the sum of the dropped squares, and the singular-vector factors that build it are not unique

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Carl Eckart, Gale Young (1936), Psychometrika 1(3):211–218 — DOI-10.1007/BF02288367

Standing in the record

Registered on 2026-09-29 at the owner’s request, because the owner’s prior-art novelty map (a private working document on concepts as projectors, the linear representation hypothesis and their neighbours) cites it: row 10: owner of the unique-factorization/low-rank machinery the Universal Convergence argument leans on. It is seeded from metadata and published descriptions and has not been read here.

It was filed Deferred, unread. NOTE-301 is the close reading of 2026-09-29, and it placed the work: Active — worth reading as the short, readable origin of the Frobenius-norm low-rank approximation theorem in the form still used. The proof is a first-order stationarity argument plus an unproved simultaneous-diagonalisation theorem, and it covers only the least-squares norm. For the general unitarily-invariant-norm statement, cite Mirsky (1960, not in the record).

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