Talk:Row echelon form
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My Precalculus book reports that Row-Echelon form contains the requirement:
"A matrix in row-echelon form has the following properties ... 2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (Called a leading 1) ..." - PRECALCULUS 7th Edition, Larson & Hostetler
Which is contrary to what is reported in the article.
Which form is correct? Serialized 00:23, 2 December 2006 (UTC)
- Apparently, there isn't universal agreement about this. Some books include that requirement and others don't. A book I have also says this, but someone else here has a book that doesn't say it. See Talk:Gaussian elimination#REF and RREF requirements. Eric119 02:52, 2 December 2006 (UTC)
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- I added a note; I think that the article should mention that there are two slightly differing definitions. -- Jitse Niesen (talk) 13:52, 2 December 2006 (UTC)
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[edit] TI-89 Example
Does anyone else think that having the example of "doing" REF on a TI-89 is not appropriate here? There are many different models of calculator and there is no need to single this one out. Such information is more appropriate for the calculator's manual. If we did include it, it would belong in Gaussian elimination, not here. Eric119 05:42, 19 December 2006 (UTC)
- I agree. Besides many calculators, there are also numerical libtaries and computer algebra systems. It makes little sense to explain how to find the REF in all these environments. Hence, I removed the section. -- Jitse Niesen (talk) 11:36, 19 December 2006 (UTC)
[edit] Excess requirement
The article read as follows before I edited it (requirements for RREF):
- All nonzero rows are above any rows of all zeroes.
- The leading coefficient of a row is always to the right of the leading coefficient of the row above it.
- All entries below a leading coefficient, if any, are zeroes.
However, this last requirement is redundant. Take the leading coefficient of any non-zero row. The elements directly below this are either:
- In a zero row, in which case the element is zero, or
- In a nonzero row, in which case that row's leading element is to the right and so the element directly below is also zero.
Thus the third requirement of the above is redundant; it results from the first two.
Unless I've screwed up.
Rawling4851 22:31, 20 January 2007 (UTC)
[edit] Triangle Matrix
What is the relationship between upper triangular matrices and matrices in row echelon form? For example, is the upper triangular matrix a special case of row echelon form? It would seem that the only requirement for a upper triangular matrix above that of row echelon form is that it be square. Is it accurate to say that all upper triangular matrices are in row echelon form? Jebix 22:01, 29 July 2007 (UTC)
[edit] Leading coefficent
The article on "leading coefficient" is not completely clear : it should be precised that the leading coefficent is only defined for non-zero rows. Striclty speaking, this precision should also appear in your first definition :
- either the matrix is the null matrix
- or the non-zero rows are all above the (eventual) zero rows, and the leading coefficient of a non-zero row which is not the first one is stricly to the right of the leading coefficent of the row above it.
--Zebulon64 (talk) 14:29, 7 March 2008 (UTC)
[edit] The Hermite Matrix
The definition of row-reduced form is a bit confusing. Here I address the comments of both Rawling and Zebulong64 above, suggest criteria to use in defining a row-reduced matrix, and correct the definition of an Hermite matrix. It would be nice, in applied mathematics, to motivate calculating the Hermite matrix H by noting that non-zero columns of I-H form a basic, independent, solution set of Ax=0. The algorithm chosen, such as Gauss-Jordan, would determine the definition of a row-reduced matrix R. The construction of H, matrix R appended with zero rows to make a square matrix, should illustrate three things: (1) From its construction, it is row-equivalent to A, so it has the same solution set. (2) From its construction, H is idempotent; that is, HH = H. Consequently, A(I-H) = H(1-H) = H - HH = H - H = 0 Consequently, a (basis for the) solution set is I-H. (3) From its construction, each non-zero column has at most c+1 non-zero elements, making the solution 'basic'. This, for example, is an interpretation of the balancing of chemical reactions by constructing an Hermite matrix, each column of A being a chemical species, each row of A being a compositional component. 'Basic' solutions have the advantage here of satisfying Gibbs's phase rule, which guarantees their interpretation as chemical reactions. The details depend upon the choice of algorithm (described in pseudocode). Geologist (talk) 01:34, 15 March 2008 (UTC)
