Elimination and Factorization
Abstract
If a matrix $A$ has rank $r$, then its row echelon form (from elimination) contains the identity matrix in its first $r$ independent columns. How do we \emph{interpret the matrix} $F$ that appears in the remaining columns of that echelon form\,? $F$ multiplies those first $r$ independent columns of $A$ to give its $n-r$ dependent columns. Then $F$ reveals bases for the row space and the nullspace of the original matrix $A$. And $F$ is the key to the column-row factorization $\boldsymbol{A}=\boldsymbol{CR}$.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2023
- DOI:
- 10.48550/arXiv.2304.02659
- arXiv:
- arXiv:2304.02659
- Bibcode:
- 2023arXiv230402659S
- Keywords:
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- Mathematics - Numerical Analysis
- E-Print:
- 5 pages, no figures, 4 references