SIAM Review, Volume 64, Issue 1, Page 181-190, February 2022.
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.

中文翻译：

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LU 和 CR 消除

SIAM 评论，第 64 卷，第 1 期，第 181-190 页，2022
年 2 月。简化的行梯形形式 ${\bf rref}(A)$ 传统上用于课堂示例：具有整数条目和低位的小矩阵 $A$排名$r$。本文创建了一个列-行秩揭示因子分解${A=CR}$，其中$C$ 中$A$ 的前$r$ 个独立列和${\bf rref}( A)$ 中的 $R$。我们希望通过帮助学生查看 $A$ 的独立列以及排名和列空间来重新构想线性代数课程的开始。如果 $B$ 包含 $A$ 的前 $r$ 独立行，则 ${A=CR}$ 的那些行产生 ${B=WR}$。$r$ x $r$ 矩阵$W$ 具有满秩$r$，其中$B$ 满足$C$。然后三重分解 ${A=CW^{-1}B}$ 以相同的方式处理 $A$（$C$ 和 $B$）的列和行。