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Journal of Convex Analysis 28 (2021), No. 2, 725--728
Copyright Heldermann Verlag 2021

The Column-Row Factorization of a Matrix

Gilbert Strang
Department of Mathematics, MIT, Cambridge, MA 02139, U.S.A.


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

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