LU and CR Elimination

Abstract

The reduced row echelon form ${\bf rref}(A)$ has traditionally been used for classroom examples: small matrices $A$ with integer entries and low rank $r$. This paper creates a column-row rank-revealing factorization ${A=CR}$, with the first $r$ independent columns of $A$ in $C$ and the $r$ nonzero rows of ${\bf rref}(A)$ in $R$. We want to reimagine the start of a linear algebra course by helping students to see the independent columns of $A$ and the rank and the column space. If $B$ contains the first $r$ independent rows of $A$, then those rows of ${A=CR}$ produce ${B=WR}$. The $r$ by $r$ matrix $W$ has full rank $r$, where $B$ meets $C$. Then the triple factorization ${A=CW^{-1}B}$ treats columns and rows of $A$ ($C$ and $B$) in the same way.