Talk:Row echelon form

Mathematics Stub‑class Mid‑priority

	Mathematics portal This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics
Stub	This article has been rated as Stub-class on Wikipedia's content assessment scale.
Mid	This article has been rated as Mid-priority on the project's priority scale.

Goto??

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

What's wrong with it? It provides a simple way of simultaneously exiting the while loop and skipping the for loop. Rgrant222 (talk) 05:57, 26 January 2011 (UTC)[reply]

Ambiguity in algorithm

the link to pseudocode does not define how "stop" should work. Does it stop the current loop? Does it stop the whole function? Is it like a "return" in C, or like a "break"?

function ToReducedRowEchelonForm(Matrix M) is
    lead := 0
    rowCount := the number of rows in M
    columnCount := the number of columns in M
    for 0 ≤ r < rowCount do
        if columnCount ≤ lead then
            stop
        end if
        i = r
        while M[i, lead] = 0 do
            i = i + 1
            if rowCount = i then
                i = r
                lead = lead + 1
                if columnCount = lead then
                    stop
                end if
            end if
        end while

Pseudocode algorithm fails

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)[reply]

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. 68.93.203.30 (talk) 14:08, 13 May 2011 (UTC)[reply]

—Preceding unsigned comment added by 190.152.74.4 (talk) 01:00, 3 June 2009 (UTC)[reply]

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) [reply]

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)[reply]

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)[reply]

This requirement is congruent with what is currently given. 68.93.203.30 (talk) 14:12, 15 April 2011 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Yes (Michael Kemp)

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)[reply]

A separate issue:

In the current version, leading coefficients have to be 1.

This need not be the case (eg see planet math). I realise that some authors (eg Anton) give the definition as currently given in this article. However, by relaxing the condition backwards substitution is still straight forward which is the point of this form anyway. —Preceding unsigned comment added by 137.166.4.130 (talk) 04:25, 4 June 2010 (UTC)[reply]

--Michael Kemp

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)[reply]

New Algorithm for Obtaining Row-Echelon Form (Not Reduced)

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)[reply]

Why is "echelon form" a wikipedia subject?

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)[reply]

Equal Sign and Assignment Sign ambiguity

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)[reply]