The Column-Row Factorization of a Matrix

The active ideas in linear algebra are often expressed by matrix factorizations :

for symmetric matrices (the spectral theorem) and

A=U\Sigma V^{\mathrm{T}} A = U Σ V T

for all matrices (singular value decomposition). Far back near the beginning comes

A=LU A = L U

for successful elimination : Lower triangular times upper triangular. This paper is one step earlier, with bases in

A=CR A = C R

for the column space and row space of any matrix – and a proof that column rank = row rank. The echelon form of

A A

and the pseudoinverse

A^+ A +

appear naturally. The “proofs” are mostly “observations”.