The Column-Row Factorization of a Matrix

Abstract

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''.

Contact details are reproduced from the original publication and may be historical.

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

15A23

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.