Geodesic
The Column-Row Factorization of a Matrix
Journal of convex analysis, Tome 28 (2021) no. 2, pp. 725-728
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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''.
Classification : 15A23
Mots-clés : Matrix, factorizations, rank, echelon form 
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     author = {G. Strang},
     title = {The {Column-Row} {Factorization} of a {Matrix}},
     journal = {Journal of convex analysis},
     pages = {725--728},
     year = {2021},
     volume = {28},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2021_28_2_JCA_2021_28_2_a22/}
}
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G. Strang. The Column-Row Factorization of a Matrix. Journal of convex analysis, Tome 28 (2021) no. 2, pp. 725-728. http://geodesic.mathdoc.fr/item/JCA_2021_28_2_JCA_2021_28_2_a22/