Johnson (1963) On a Theorem Stated by Eckart and Young

Summary

Johnson supplies a general proof of the matrix-decomposition theorem that Eckart and Young stated but did not prove in their 1936 paper — the result underlying their low-rank approximation solution. The theorem asserts that for any real matrix $A$ there exist orthogonal matrices $P$ and $Q$ such that $P'AQ$ is a real diagonal matrix with no negative elements (the singular value decomposition, called the "basic structure" by Horst). Eckart and Young had directed readers to references that either lacked the proof or (MacDuffee) covered only the square nonsingular case; Johnson fills the gap with a proof valid for general real matrices.

Key Claims

• The stated-but-unproved theorem. Eckart-Young's low-rank result rests on the claim that any real matrix reduces, via orthogonal $P,Q$, to a nonnegative real diagonal matrix $P'AQ$ — i.e., that the SVD ("basic structure") exists.
• The gap in 1936. Eckart and Young cited two references for the proof; one does not contain it, and the other (MacDuffee) proves only the square, nonsingular case (Sylvester gives a more complicated square-case discussion).
• Johnson's contribution. A general proof of the theorem for arbitrary real (including rectangular, singular) matrices, securing the foundation of the Eckart-Young solution that had become a cornerstone of factor analysis and related multivariate techniques.

Concepts Introduced or Extended

• Low-Rank Approximation

Entities Mentioned

• Richard M. Johnson
• Carl Eckart
• Gale Young

Quotes

"Proof is given for a theorem stated but not proved by Eckart and Young in 1936, which has assumed considerable importance in the theory of lower-rank approximations to matrices, particularly in factor analysis."

My Take

A small but genuine piece of scholarly hygiene: the Eckart-Young low-rank theorem had become load-bearing across psychometrics and multivariate statistics while its underlying SVD-existence claim sat formally unproved in the general case for nearly three decades. Johnson closing that gap is a reminder that "everyone knows it's true" and "it has been proved" are different states. For the wiki it is a footnote to Low-Rank Approximation rather than a new idea, but a useful one — and a nice bit of trivia that the proof came out of Procter & Gamble's market-research group (Johnson later pioneered conjoint analysis).