The Approximation of One Matrix by Another of Lower Rank

1936/09/01 by Carl Eckart, Gale Young · 3,824 citations
Computer Science · Mathematics · #Algebra over a field #Applied mathematics #Calculus (dental) #Combinatorics #Computer science #Interpretation (philosophy) #Low-rank approximation #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #Pure mathematics #Rank (graph theory)

paper · doi:10.1007/bf02288367

published in Psychometrika 1(3), 211-218 (Springer Science+Business Media)

openalex publication_date 1936/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution. A hypothetical interpretation of the canonic components of a score matrix is discussed.

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