Defactoring: Difference between revisions

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There are several well-known echelon forms for matrices that predate DCF. Let's review them and their properties.

The most general form, with the fewest constraints, is simply called '''[https://en.wikipedia.org/wiki/Row_echelon_form Row Echelon Form]''', or '''REF'''. Its only constraint is ''echelon<ref>The name "echelon" is a French word for a military troop formation with a similar triangular shape: https://en.wikipedia.org/wiki/Echelon_formation.</ref> form'': each row's pivot, or first nonzero entry, is strictly to the right of the preceding row's pivot. This single constraint is fairly weak, and therefore REF does not produce a unique representation. This constraint is shared by every matrix form discussed here.<ref>Note that the definition of REF is inconsistent and sometimes it includes some of the constraints of RREF, discussed further below. See: https://www.statisticshowto.com/matrices-and-matrix-algebra/reduced-row-echelon-form-2/</ref>

The most general form, with the fewest constraints, is simply called '''[https://en.wikipedia.org/wiki/Row_echelon_form Row Echelon Form]''', or '''REF'''. Its only constraint is ''echelon<ref>The name "echelon" is a French word for a military troop formation with a similar triangular shape: https://en.wikipedia.org/wiki/Echelon_formation.</ref> form'': each row's pivot, or first nonzero entry, is strictly to the right of the preceding row's pivot. This single constraint is fairly weak, and therefore REF does not produce a unique representation. This constraint is shared by every matrix form discussed here.<ref>Note that the definition of REF is inconsistent and sometimes it includes some of the constraints of RREF, discussed further below. See: https://www.statisticshowto.com/matrices-and-matrix-algebra/reduced-row-echelon-form-2/</ref><ref>REF also requires that all rows that are entirely zeros should appear at the bottom of the matrix. However this rule is only relevant for rank-deficient matrices. We'll be assuming all matrices here are full-rank, so we don't have to worry about this.</ref>

In the below example, <span><math>x_{ij}</math></span> represents any number.

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 There are several well-known echelon forms for matrices that predate DCF. Let's review them and their properties.
-The most general form, with the fewest constraints, is simply called '''[https://en.wikipedia.org/wiki/Row_echelon_form Row Echelon Form]''', or '''REF'''. Its only constraint is ''echelon<ref>The name "echelon" is a French word for a military troop formation with a similar triangular shape: https://en.wikipedia.org/wiki/Echelon_formation.</ref> form'': each row's pivot, or first nonzero entry, is strictly to the right of the preceding row's pivot. This single constraint is fairly weak, and therefore REF does not produce a unique representation. This constraint is shared by every matrix form discussed here.<ref>Note that the definition of REF is inconsistent and sometimes it includes some of the constraints of RREF, discussed further below. See: https://www.statisticshowto.com/matrices-and-matrix-algebra/reduced-row-echelon-form-2/</ref>
+The most general form, with the fewest constraints, is simply called '''[https://en.wikipedia.org/wiki/Row_echelon_form Row Echelon Form]''', or '''REF'''. Its only constraint is ''echelon<ref>The name "echelon" is a French word for a military troop formation with a similar triangular shape: https://en.wikipedia.org/wiki/Echelon_formation.</ref> form'': each row's pivot, or first nonzero entry, is strictly to the right of the preceding row's pivot. This single constraint is fairly weak, and therefore REF does not produce a unique representation. This constraint is shared by every matrix form discussed here.<ref>Note that the definition of REF is inconsistent and sometimes it includes some of the constraints of RREF, discussed further below. See: https://www.statisticshowto.com/matrices-and-matrix-algebra/reduced-row-echelon-form-2/</ref><ref>REF also requires that all rows that are entirely zeros should appear at the bottom of the matrix. However this rule is only relevant for rank-deficient matrices. We'll be assuming all matrices here are full-rank, so we don't have to worry about this.</ref>
 In the below example, <span><math>x_{ij}</math></span> represents any number.