Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 28 (2021), No. 2, 725--728Copyright Heldermann Verlag 2021 The Column-Row Factorization of a Matrix Gilbert Strang Department of Mathematics, MIT, Cambridge, MA 02139, U.S.A. gilstrang@gmail.com [Abstract-pdf] The active ideas in linear algebra are often expressed by matrix factorizations\,: $S=Q\Lambda Q^{\mathrm{T}}$ for symmetric matrices (the spectral theorem) and $A=U\Sigma V^{\mathrm{T}}$ for all matrices (singular value decomposition). Far back near the beginning comes $A=LU$ for successful elimination\,: Lower triangular times upper triangular. This paper is one step earlier, with bases in $A=CR$ for the column space and row space of any matrix -- and a proof that column rank = row rank. The echelon form of $A$ and the pseudoinverse $A^+$ appear naturally. The proofs'' are mostly observations''. Keywords: Matrix, factorizations, rank, echelon form. MSC: 15A23 [ Fulltext-pdf  (68  KB)] for subscribers only.