Talk:Row echelon form

message: Looking at the second pseudo code module, what is with the goto statement.
My IQ >> 160 (talk) 22:16, 20 December 2010 (UTC)
thread_title:  Goto?? 

message: For this matrix:
{{0 0 0 0} {0 -1 2 -5} {1 2 -4 8} {5 0 -3 2}}
The first pseudocode algorithm failed for several matrices for me me as well.  I suggest removing the code entirely.
: Right, the code is bad.  For example there is no clear exit condition for the first while loop if it only sees zeros, unless stop function means something non-obvious.  We can't accept stop function anyway since it is not a standard pseudocode component.  What about  this, does it work? McKay (talk) 23:53, 29 March 2011 (UTC)
replies: text:  McKay - I haven't had a chance to check the Gaussian elimination pseudocode myself, however, I do have a working implementation that I made based upon my the aforementioned erroneous algorithm (which has since been removed from the page).  Although it is a C implementation, I suppose that it would suffice as an example if I remove some of the particularly C specific parts of the code.  In my opinion, even the current pseudocode for row echelon form looks C-like due to the starting at zero for each of the matrix indices, so I am strongly considering including C based pseudocode if there is any interest from the community.  I am still relatively new to the editing guidelines on pseudocode, so please advise. KlappCK (talk) 14:09, 13 May 2011 (UTC)

text: 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)
replies: text: 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)

text: This requirement is congruent with what is currently given. KlappCK (talk) 14:10, 13 May 2011 (UTC)
thread_title:  Pseudocode algorithm fails 

message: 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)
replies: text: 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)

text: I imagine that the push for TI-89 implementations is due to prevalence TI-83 and TI-89 in many American mathematics class rooms.  Furthermore, TI-8* code looks a lot like pseudocode.  That said, I am certainly stating my own opinion here, not that of Wikipedians as whole. Klappck (talk) 17:38, 12 April 2011 (UTC)
thread_title: TI-89 Example

message: 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)
thread_title:  Excess requirement 

message: 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)
thread_title:  Triangle Matrix 

message: 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)
thread_title: The Hermite Matrix

message: I've posted a psuedocode algorithm for converting a matrix to it's row-echelon form. Hopefully the algorithm stands the test of time better than the first (for reduced row-echelon form). I've left the first in place in the hopes that someone is clever enough to alter it, rather than completely rewrite it, so that it works. Hope this helps.  
Rgrant222 (talk) 04:50, 13 September 2010 (UTC)
replies: text: Rgrant222 - I think that all that needs to be done to turn the current pseudocode algorithm for row echelon form to the reduced variety is to add a division of the pivot row by the pivot element before that rows values are subtracted from the others to make all other elements in the current pivot column equal to zero.  In theory, that means that one properly placed for loop should do the trick.KlappCK (talk) 14:13, 13 May 2011 (UTC)
thread_title:  New Algorithm for Obtaining Row-Echelon Form (Not Reduced) 

message: I do not understand why wikipedia should elevate the terminology "echelon form" to such importance that it is a separate page.  No links are given to other subjects that use the terminology except Gaussian elimination.  It suffices for the Gaussian elimination pages to point out that a triangular matrix is sometimes described as being in "echelon form."  In fact, the echelon terminology is not used in numerical analysis, which is the principal field that studies Gaussian elimination. Jfgrcar (talk) 17:07, 2 October 2010 (UTC)
thread_title:  Why is "echelon form" a wikipedia subject? 

message: In programming there are two types or equal signs: one indicating assignment, for example assign a value to a variable (i:=i+1) and other verifying equality, for example used in conditional sentences (if a=b then a:=2*b). In the pseudo code there is inconsistence between these differences.  — Preceding unsigned comment added by Zeusescudero (talk • contribs) 01:45, 9 March 2011 (UTC)
thread_title:  Equal Sign and Assignment Sign ambiguity 

message: I have a linear algebra textbook (Linear Algebra with applications, 8 ed. by Steven J Leon, ISBN-13: 987-0-13-600929-0) that lists that a matrix is considered in row echelon form only if "the first nonzero entry in each nonzero row is 1". (page 13) The other requirements it lists are consistent with this article, and it gives examples (with answers) that confirm its definition and contradict this article's definition.
Is the textbook's interpretation invalid (and if so, what information and sources could I send the publisher to rectify this in subsequent editions), or does it represent a valid alternate interpretation? Is this alternate interpretation notable enough to justify mentioning in the article? (It's certainly the one I must use to pass MATH 301 at Boise State University, and any other institution that uses this book.)  — Preceding unsigned comment added by 75.174.125.203 (talk) 23:42, 5 September 2013 (UTC)
thread_title:  First nonzero entry in each row = 1  (left to right) 

message: - The relationship between row echelon form and upper triangular matrices.
As an example: \begin{pmatrix}0&1&1\\0&1&2\\0&0&0\end{pmatrix} is an upper triangular matrix that is not in row echelon form.
 - Most matrix decomposition algorithms produce an  R that is in row echelon form.
Most matrix decomposition algorithms  A \mapsto M\cdot R \quad\text{so that}\quad A = MR  where  R is an upper triangular matrix actually output an  R that is in row echelon form (most notably this is the case for LR-decomposition). This is especially important for forward/backward substitution (e.g. for above example triangular matrix, forward substitution could not be applied). In the current form, forward/backward substitution is expanded upon on the site for triangular matrices, where it neglects the case that l_{i,i} is zero.
 - Forward/backward substitution should be on this site, not on the site for triangular matrices
 - Reduced row echelon form gives an easy method to describe the vector space spanned by an linear equation system  Ax=b.
As an example, take 
\left[ \begin{array}{ccccc}
1 & 0 & a_1 & 0 & b_1 \\
0 & 1 & a_2 & 0 & b_2 \\
0 & 0 & 0 & 1 & b_3
\end{array} \right]\cdot\left[ \begin{array}{c} x_1\\x_2\\x_3\\x_4\\x_5\end{array}\right]
= \left[\begin{array}{c} y_1\\y_2\\y_3\end{array}\right]
.
Then we can solve the first row for x_1, the second for x_2 and the third for x_4, and take the remaining variables as free variables. We therefore arrive at the solution set
 \left\{
\left[ \begin{array}{c} y_1-a_1 x_3-b_1 x_5\\y_2-a_2 x_3 - b_2 x_5\\x_3\\y_3-b_3 x_5\\x_5\end{array}\right]\mid x_3,x_5\in\mathbb{R}\right\}
This method is described e.g. in the German book "Lineare Algebra by Jörg Liesen" in Chapter 6, "Lineare Gleichungssysteme"
(This method basically is a generalized backward substitution that works for underdetermined systems and returns all solutions instead of one)
Sanitiy (talk) 17:31, 21 December 2019 (UTC)
thread_title:  Additional topics that could be added to this article 

message: In adding citations to the article, I noticed that “pivot” is given as a synonym of “leading coefficient.” But I have been unable to find this sense of “pivot,” instead only finding the sense I am more familiar with—i.e. a leading element selected during Gaussian elimination (or perhaps some other algorithm) (See Pivot element). So the term is meaningless when not referring to a specific algorithm and even then not every leading coefficient would be a pivot. Of course, I have limited time and textbooks to look through and I wouldn't be astonished if pivot had taken on the meaning given in the main article. But I think it is more likely that it was mistakenly added and overlooked by other editors.
Bhbuehler (talk) 00:31, 17 December 2022 (UTC)
replies: text: I agree. Moreover, even if some authors have used "pivot" in this sense, this adds nothing to the article to mention this here, as this meaning belongs to Pivot element, and more sepcifically the its section Pivot position. So, I have rewritten the sentence, and removed also "leading coefficient" (the use of "coefficient" instead of "entry" seems uncommon for native English speakers, and its use in English seems result from mimicking non-English languages such as French, where the equivalent of "entry" is not used) D.Lazard (talk) 09:24, 17 December 2022 (UTC)
thread_title:  Improper pivot definition