Talk:Determinant

Right handed coordinante

the following sentence is not clear. "The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed." what does "right-handed coordinate system" means? the "coordinate system" article does not mention it. amit man

archive

I've set up an automated archiving for this talk page. Jakob.scholbach (talk) 20:43, 27 March 2011 (UTC)

Possible to do/see also items

linear algebra/analytic geometry

linear independence/collinearity, Gram determinant, tensor, positive definite matrix (Sylvester's criterion), defining a plane, Line-line intersection, Cayley–Hamilton_theorem, cross product, Matrix representation of conic sections, adjugate matrix, similar matrix have same det (Similarity invariance), Cauchy–Binet formula, Trilinear_coordinates, Trace diagram, Pfaffian

types of matrices

special linear group, special orthogonal group, special unitary group, indefinite special orthogonal group, modular group, unimodular matrix,

number theory/algebra

Pell's equation/continued fraction?, discriminant, Minkowski's theorem/lattice, Partition_(number_theory), resultant, field norm, Dirichlet's_unit_theorem, discriminant of an algebraic number field

geometry, analysis

conformal map?, Gauss curvature, orientability, Integration by substitution, Wronskian, invariant theory, Monge–Ampère equation, Brascamp–Lieb_inequality, Liouville's formula, absolute value of cx numbers and quaternions (see 3-sphere), distance geometry (Cayley–Menger determinant), Delaunay_triangulation

open questions

Jacobian conjecture, Hadamard's maximal determinant problem

algorithms

polar decomposition, QR decomposition, Dodgson_condensation, Matrix_determinant_lemma, eigendecomposition a few papers: Monte carlo for sparse matrices, approximation of det of large matrices, The Permutation Algorithm for Non-Sparse Matrix Determinant in Symbolic Computation, DETERMINANT APPROXIMATIONS

examples

reflection matrix, Rotation matrix, Vandermonde matrix, Circulant matrix, Hessian matrix (Blob_detection#The_determinant_of_the_Hessian), block matrix, Gram determinant, Elementary_matrix, Orr–Sommerfeld_equation, det of Cartan matrix

generalizations

Hyperdeterminant, Quasideterminant, Continuant (mathematics), Immanant of a matrix, permanent, Pseudo-determinant, det's of infinite matrices / regularized det / functional determinant (see also operator theory), Fredholm determinant, superdeterminant

other

Determinantal point process, Kirchhoff's theorem,

books

[1]

Opening sentence

Is "special" a mathematical term outside my experience? If so, should there be an article about it, with a hyperlink. Otherwise, what is its force in this sentence. In any case, about 20 books I have at hand define a determinant as a SQUARE ARRAY with the value given by the sum of products formula, NOT as a number. If a determinant is defined as a number, how can all the manipulations on rows and columns be discussed? They have been lost in the evaluation. The rest of the lede certainly gives prominence to topics that I have never seen mentioned in an introduction to the topic. I have to stop here -- but the rest of the article is rather disjoint from customary accounts. Might it be a good idea for someone to go through some standard textbook discussions and follow standard coverage? Michael P. Barnett (talk) 02:50, 1 May 2011 (UTC)

The determinant is a number, not an array. It is a property of the array, and row and column operations on the array will affect the value of the determinant, but the determinant itself is a number (a member of the underlying field or ring). The lede seems reasonable to me. I agree that the article feels a bit disjointed, the result of many hands over time, and it could used a general edit to integrate the pieces with a common style; but the content covers the usual standard topics. There is plenty more that could be added, as noted in the section above, though whether all that material belongs in one article is perhaps something that should be discussed. Anything beyond that you would like to add? -- Elphion (talk) 13:50, 1 May 2011 (UTC)

Trying to understand how your many books speak of the determinant as an array, I should mention that there is the usual conflation of the operator and the things it produces. For example, the absolute value of a real number is real number; but there is also the absolute value function on the reals that returns the absolute value of the argument. Similarly, the determinant of a matrix is a number, but there is also the determinant operator that takes arrays into the underlying ring of numbers. -- Elphion (talk) 15:05, 1 May 2011 (UTC)

Many thanks for your comment. I was posting a further comment that hit an edit conflict with yours. Herewith a slight amendation of what I had written. I assume that you have formal training in mathematics. My degrees are in chemistry, but I have picked up some awareness of mathematical topics over the years. Since posting my comment last night, I had checked further in books I have at hand, and found the definition as a number in authoritative sources -- Rektorys, Itô, and others. Unfortunately I had gone first to Mathematics of Physics and Chemistry that I used to teach myself about determinants in 1948+. Sorry. Also I have got into the habit of representing the array that defines a determinant as det[...] in Mathematica sessions, replacing det by Det when I want automatic evaluation (which is seldom).

I got into the matrix element and determinant articles yesterday because I could not remember whether a matrix is Hermitian if or iff it is self-adjoint, and was feeling a bit bruised by WK articles beyond my understanding. It would be presumptuous for me at this time to suggest significant changes to the lede for Determinants that has been gone over by people with far more mathematical knowledge than me. But I hope it is not presumptuous if I mention how I think it might have been started, had nothing been written already. There was an article in the Guardian Weekly last week that reported concern by WK that so few academics contribute to WK articles in their respective professional fields. Abstract algebra is NOT mine, but I have taught math literacy extensively, to students with a wide range of comprehension (including graduate humanists -- answering their questions was extremely beneficial to my understanding).

Back to the lede

1. Suppose someone, Pat Doe, who can handle elementary algebraic manipulation, but has never encountered sets and is afraid of geometry, wants to learn about determinants. I assume that the opening sentences of a lede should be written in a way that avoids possible misunderstanding by someone working in isolation, without a tutor to turn to. I have looked at the present lede with an eye to how it might be misunderstood. Could the dimensionality 2 be considered "an important number associated with any square matrix"? Could the rank be considered an important number associated with it? The trouble is with the word "any". If you have only taught elite students, you may not realize how things that are obvious to you are not necessarily obvious to others. I belong to the "ability to misunderstand" contingent, and empathize.

2. The rest of the opening paragraph will put off a Pat Doe who follows the links, which require more and more mathematical background.

3. The article does seem a bit slanted to the geometrical, relative to all the material I have at hand. But my personal library is a bit outdated, and I cannot get to a major library for several days.

4. Hence my following effort, which I recognize will go no further, but can do no harm.

Michael P. Barnett (talk) 15:52, 1 May 2011 (UTC)

(outdent) For clarity I've taken the liberty of duplicating your signature from below to apply to the remarks above, since the intervening section head guarantees that they will get separated.

I sympathize entirely with your bruising experience in the math articles. Unfortunately, the professional touch often contributes impenetrability rather than lucidity. There are several problems here. In the first place, WP is an encyclopedia, not a text book, and striking the proper balance in presentation is not easy. EB has this problem as well; several of their technical articles jump right in to very technical discussion. And professional mathematicians will overlook things that need explaining: your mention of simultaneous equations below is a glaring omission from the current lede (especially given the history of determinants), because every mathematician will "just know" that that's included in "linear algebra".

In WP the problem is compounded by frequent editing by different hands: the undergraduate throws in a favorite example, and the professional decides the discussion is not suitably general. Both are often useful additions, but without care the organization and readability of the article will suffer. It is hard and time consuming to present technically involved concepts lucidly in the first place, and the will and energy to keep them lucid over time can be hard to maintain. And on taking a closer look at the current article, I will revise my assessment above: it's more than "a bit" disjointed; it needs a thorough overhaul.

There are some good points in your suggested lede below, but I have some reservations, appended below.

-- Elphion (talk) 17:53, 1 May 2011 (UTC)

An alternative opening

In algebra, the determinant of a square matrix (that is, a square array) of numbers is a single number that is computed from the elements of the array by a simple rule that the following example illustrates. The determinant: ${\begin{vmatrix}1&2\\3&4\end{vmatrix}}\$ has the value 1×4-2×3=-2. The rule converts a 3×3 determinant to the sum of 3!=6 terms. Each of these is the product of 3 elements of the matrix. In general, the value of an n×n determinant consists of n! terms. Each of these is the product of n elements of the matrix, selected in a special way. Many kinds of mathematical expression can be used as elements of a matrix, and the corresponding determinants are computed by the rules that are used when the elements are numbers. Determinants are used in calculations throughout engineering and the natural and social sciences. They are particularly important in the solution of simultaneous equations. Determinants have many properties that simplify their computation, which becomes enormously time consuming when the determinants are large. The properties of certain determinants can be explained geometrically. Determinants that consist of other kinds of mathematical objects are used in advanced mathematical and scientific theories.

(further paragraphs appropriate to successive sections of article) Michael P. Barnett (talk) 15:52, 1 May 2011 (UTC)

A good start. Some reservations:

1. The "simple" rule is simple only when n < 3. I think it's a mistake to get into computation (even by example) in the lede. I would like to see a separate section in the article on computation, clearly labeled "Computation of determinants", and the n! number of terms should be explicitly mentioned there.

2. "Many kinds of mathematical expression can be used as elements of a matrix, and the corresponding determinants are computed by the rules that are used when the elements are numbers." I see what you're getting at, but it needs to be stated more carefully. (One could, I suppose, speak of matrices over function rings, but that's not necessarily helpful to the layman!)

3. Similarly, "Determinants that consist of other kinds of mathematical objects are used in advanced mathematical and scientific theories." -- Remarks similar to 2. Also, "advanced mathematics and science" suffices, without the theories.

4. "The properties of certain determinants can be explained geometrically" doesn't really convey much without an example. I think the example currently in the lede might as well stay (and it's a good example); otherwise the geometric aspect probably ought to be eliminated from the lede.

-- Elphion (talk) 17:53, 1 May 2011 (UTC)

The new lede and general issues it raises

Many thanks for rewriting the lede the way you have. It has given me fresh insights to a number of matters. But I wonder who else it can benefit. Which leads me ask, for whom are the mathematics articles in WK intended -- readers or Editors and, if readers, with what range of knowledge? Should the WK article on determinants be readable by someone who can understand explanations of determinants in introductory texts and in reference works that are in many public libraries?

The lede is now very different from the opening of articles about determinants in every published encyclopedic work on mathematics and from every standard textbook and monograph that deals in all or in part with linear algebra that I have inspected regarding the topic. Some are introductory, some are advanced. Perhaps the most institutionally authoritative is the Encyclopedic Dictionary of Mathematics of the Mathematical Society of Japan, that has forewards by the President of that society and the President of the American Mathematical Society at the time of publication. I can provide a full list.

Of all these works, only the text by Poole mentions the geometrical interpretation, and that is in the context of the customary treatment, and diluted by numerous conventional examples.

I can provide simple statistics on the vast number of papers that use determinants because of their non-geometrical properties. What is the evidence for correspondingly massive use of geometrical properties?

I recognize that a strength of WK is potential for updates that provide currency. But does putting the geometry first gibe with widespread current literature and teaching? Is there any need for verifiability that a non traditional approach has been adopted to a major extent, to justify mentioning it first in the lede?

Scholarly papers published in peer reviewed journals often give references to sources of background information, often citing two or three that are typical and consistent. It would be helpful for people increasingly dependent on internet access for WK articles to conform. I could not do this with the Determinant article or with many other WK articles on math and science because they are so inconsistent with outside world presentations.

The impact of the opening sentence of a lede is extremely important. Someone with knowledge of determinants, seeking to refresh or check some details, might well give up after the opening paragraph, because it is so disjoint with what they remember.

Someone who is a bit more tenuous and who is perplexed, as I was, because they do not realize that the word "volume" is being used in the abstract n-dimensional sense, and read on, might become even more concerned. Just follow the link to measure. Then check the following statements.

1 (opening of second paragraph) "To qualify as a measure (see Definition below), a function that assigns a non-negative real number or +∞ to a set's subsets must satisfy a few conditions. One important condition is countable additivity."

A reader who has absorbed the comment in Determinant that sign shows orientation might give up at this point and assume the author of the new Determinant lede is abusing terminology.

However, a reader who goes on will find, under Measure (mathematics)#Generalizations:

2. "For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity."

So why the restriction in the first place, and what needs to be satisfied for this generalization to be safe (in sense of not opening a route to fallacies)? The reader might give up trying to use WK math articles at that point because they seem to lack the rigour that should attend the terminology that is being used. Worse, if the reader goes on for about another 15 lines and finds

3. "The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer."

Not so OK -- a very clear indication that a WK writer disagrees with outside sources, raising concern about who to believe. And the word "mysterious" really jars in mathematical discourse that to someone with my limited knowledge seems unnecessarily eclectic. This can engender a scepticism about WK mathematics at large.

However, if the reader guessed that "measure of volume" in two dimensions means "area" (in its elementary school usage), and wants confirmation, the cursory Google search I just ran did not help.

So, for a topic that dozens of textbooks and encyclopedic works explain concisely and readably, WK just takes the time of a tenacious reader without providing help.

Am I out of order on this? If not, is the situation remediable? Michael P. Barnett (talk) 17:37, 2 May 2011 (UTC)

OK, several points. This response is not meant as negative criticism, and as I indicated above, the article does need work. Given my advice below to complain on talk pages, I think it's healthy that you're complaining on the talk page!

Just to be clear: the "new" lede is hardly new -- I only cleaned up the language a bit. I hope I haven't misled you to anticipate a major rewrite on my part any time soon; I haven't the time.

But trying to get me to agree that the geometric aspect of deteriminants is not a key point is a losing battle. This is covered it most modern introductory linear algebra texts, for the simple reason that it is key to understanding the geometry of linear transformations and their use in geometric applications like the calculus. If anything, I think this is not adquately reflected in the current article. Whether a transformation collapses the domain (det = 0) is important in analyzing simultaneous equations (and as I said above, the application to simultaneous equations does deserve explicit mention in the lede), but that's hardly the whole story. And the lede clearly states that the absolute value of the determinant deals with measure. Negative volume is never suggested.

Your complaints about the article Measure (mathematics) should be taken there, where editors can see them. That that article has its own issues is not a good reason for not linking to it, and not a good reason for rehashing the topic here for the points relevant to determinants. That way madness lies; we have to rely on links or the articles will expand beyond control.

Of course you are right that an inline judgment like "a misnomer" should be referenced. But its presence doesn't dismay me. It is a good signal that I should be wary of taking the term too literally. It would be better if the editor had explained why it's a misnomer. But the math articles are not like, say, the articles on TV series: that an editor wrote that means there's likely something to it. An appropriate response is to tag it with {{cn}}.

And it certainly doesn't bother me that this "can engender a scepticism about WK mathematics at large" -- *any* WP article should be read skeptically; nothing here should be taken on blind faith. You must evaluate the information as presented, and if you find it wanting, complain on the talk page. You seem to be asking for a grand unified approach to all the math articles, but as I implied above, such a thing is not likely to happen. The field is too vast. The best we can hope for is a collection of information that a large number of people find useful, and that gets improved when people don't find it so.

Your suggestion of listing further reading for all math articles has merit. You might want to take that to Wikipedia talk:WikiProject Mathematics.

-- Elphion (talk) 21:57, 2 May 2011 (UTC)

Thanks to you two for working on the lead. I also have this article on my todo-list (so far, didn't get much further than cleaning up some parts and creating the above sketch of possible topics). Before you engage in lengthy discussions about the lead, though, I'd like to kindly suggest to work on the article first. The lead is important, but generally changes a lot when the article changes. I often found it helpful first having a sound and stable article, then working on the lead. (Otherwise the time and energy you are spending on the lead now might not be used 100% effectively, since it might need to be reworked later anyway.)

I hope to join in once I'm done with the Logarithm FAC. Sigh... Jakob.scholbach (talk) 22:04, 2 May 2011 (UTC)

Thanks for the advice. But to clarify, I did not hope to convince anyone of anything, just wanted to test the waters of whether I should try to be responsive to recent newspaper article about the concern of WK about shortage of academics who contribute from their field of expertise and see if the perspective based on my experience could help (teaching, research, publication). I am not offended at being written off as a fossil. I am 82. Whatever does or does not go in the lede, it seems clear that NPOV as used here weighs heavily against readers who are concerned with traditional use of determinants, that references are not needed outside the history section, and readers have to know when words that are commonplace, e.g. "volume", are used with technical meaning, and words that look a bit technical, e.g. "characteristic number" do not. Michael P. Barnett (talk) 01:01, 3 May 2011 (UTC)

Welcome to WP. And yes, we can always use people with a scientific background (and also people without). The better the articles become, the more knowledge and care is needed to improve them. Hope you stay around--WP:WPM is generally a quite nice atmosphere. Again, welcome!

Observing Wiki guidelines

1. If successive sentences in the lede are supposed to indicate content of successive sections of article, a new first section should be provided or the first paragraph of present lede should be deleted. Or the present lede rearranged and a geometrical section put later in the article.

There is no strict rule on this, but surely the lead should roughly match the article. In this case, I don's see why you want to delete a part of the lead. Section 1.1. does mention the parallelogram interpretation of det of 2x2 matrices. Jakob.scholbach (talk) 09:52, 4 May 2011 (UTC)

Also there are sections "Volume" and "Jacobian determinant" under "Applications". I am personally not very happy with the emphasis on the volume interpretation because it is not very general (quite hard to give any clear meaning to it unless entries are real numbers) and in fact not very elementary either; the only reason for mentioning it early is that it is intuitive (which general determinants maybe are not). I did modify the sentence in the past (notably stressing real numbers) and am responsible (I think) for the "measure of volume" phrase, which is admittedly not very beautiful. However saying either just measure or just volume immediately leads to the question "of what?". Also it is unfortunate that volume means area in dimension 2, but this is common for geometric terminology in varying dimension ("space", "parallelepiped", "hyperplane") that is often anchored in dimension 3. Marc van Leeuwen (talk) 12:03, 4 May 2011 (UTC)

2. If a definition is supposed to have a verifiable citation attached, I think the geometrical definition should be close enough to a published definition to avoid the "original thought" that seems to have gone into the present wording, and the citation should be provided.

In the guise of the area of the parallelogram or volume of the parallelepiped, the material is standard. Of course, a reference for this cannot hurt. Jakob.scholbach (talk) 09:52, 4 May 2011 (UTC)

3. In the rest of the article, a few more citations seem necessary. Unless WK guidelines are waived for articles on math. Michael P. Barnett (talk) 21:02, 3 May 2011 (UTC)

True. In case you don't know: WP has a number of citation templates which make the references look a little more smooth and standardized. For this and more, see Wikipedia:WikiProject_Mathematics/Reference_resources. To format the {{Citation}} templates, I once wrote a database (called zeteo, see there, too). Jakob.scholbach (talk) 09:52, 4 May 2011 (UTC)

Some historical comments

Infinite determinants are of considerable importance in planetary theory aka celestial mechanics from 19th century on. It is significant that Modern Analysis by Whittaker and Watson, which marked a major development in mathematics, certainly in England, has substantive section on infinite determinants (2.8, pp 36-37). Infinite determinants with a particular systematic structure, truncated for numerical work, were the basis of the calculation of electronic structure of 2 electron atoms by Pekeris -- outside mainstream of atomic energy calculations, but extremely important at time, providing theoretical results comparable in accuracy with experimental. Quickest way for me to give references to several relevant papers is bibliography of Barnett, Decker and Krandick, J Chem Phys 114 23 10265 2011. (This introduced further computational tactics, but I am not pushing for its mention). Michael P. Barnett (talk) 21:02, 3 May 2011 (UTC)

Additive rule for determinants

I've removed the following section, recently added by 129.67.40.84 (talk). The reason is that this is clearly recent research, as far as I can see not yet published in a peer-reviewed journal, the reference is to transparencies of a presentation that in themselves do not provide sufficient context to understand the statement fully, and that the text contributed to the WP article is taken from this presentation unmodified. While the material could be interesting, I deem it premature for inclusion in the determinant article, more so because it could easily confuse readers looking for a simple formula for the determinant of a sum of two matrices (which does not exist, note the condition $N\geq n+1$ that says this only applies to sufficiently large sums of determinants). Of course it could be inserted in the article once the situation is sufficiently improved. Marc van Leeuwen (talk) 07:05, 6 May 2011 (UTC)

I inserted the section, perhaps too quickly, because the result is quite interesting, and seams to fit well into the article in general. I've tried to clean it up some, and to help people avoid the trap you rightfully mentioned. On the overall correctness of the theorem I have tested it for many values of N and n, so at least it is not obviously false. --Thomasda (talk) 17:32, 9 May 2011 (UTC)

Is this published somewhere (other than slides of a talk)? If no, we cannot take it. If yes, we can in principle take it, but even then I think we should devote at most one sentence to this theme. Before doing that, we should check whether this theorem is notable, possibly by looking for papers that cite this given paper. Jakob.scholbach (talk) 21:48, 9 May 2011 (UTC)

It's published a few places. The latest version seams to be this one at arxiv.org. I don't know how to find out if it is notable, but the WK article contains a lot of stuff on properties for calculation of determinants. As Marc said, many people probably come wondering if there is an addition formula as well. Since it is such a simple question to ask, it would be nice to have a discussion of what is possible and not possible in that regard. --Thomasda (talk) 13:29, 10 May 2011 (UTC)

By publish I did not mean preprints such as arxiv, nor that someone uploads it on some web page. We need to refer to peer-reviewed scientific journals. Is this published in any such journal? Jakob.scholbach (talk) 14:35, 10 May 2011 (UTC)

Is this result really notable on its own merits? It doesn't really qualify as a computational device: the RHS is significantly more complicated than the LHS, involving a whole boatload of matrices no smaller than the original. It would be simpler to add the matrices (trivial) and compute the determinant once. And as the discussion above indicates, this article has more pressing needs than to indicate why this result deserves space. Elphion (talk) 15:13, 10 May 2011 (UTC)

Determinant of Sums of Matrices

Given enough matrices, the determinant of their sum can be found using an Inclusion–exclusion principle like formula. More precisely, the number of matrices summed must be greater than the size of each matrix. Notice this means, that this doesn't work for $det(A_{1}+A_{2})$ unless $A_{1}$ and $A_{2}$ are 1x1 matrices.

The theorem says, that "given $A\in M_{n}$ and an integer $N$, with $N\geq n+1$. For any $N$-tuple $S=(A_{1},A_{2},...,A_{N})$, $A_{i}\in M_{n}$, $i=1,...,N$, the following relation holds:"

$\sum _{k=0}^{N}(-1)^{k}\sum _{\Omega \in \Sigma (S),|\Omega |=k}det(A+\sum _{A_{i}\in \Omega }A_{i})=0$.

Here, $|\Omega |=k$ means that $\Omega$ is a formal sum with $k$ summands, and that $A_{i}\in \Omega$ means that $A_{i}$ is a summand in $\Omega$.

For example, that theorem tells us, that for matrices of size 1x1, 2x2 or 3x3 the following equation holds:

${\begin{aligned}det(A_{1}+A_{2}+A_{3}+A_{4})&=det(A_{1}+A_{2}+A_{3})+det(A_{1}+A_{2}+A_{4})+det(A_{1}+A_{3}+A_{4})+det(A_{2}+A_{3}+A_{4})\\&-det(A_{1}+A_{2})-det(A_{1}+A_{3})-det(A_{1}+A_{4})-det(A_{2}+A_{3})-det(A_{2}+A_{4})-det(A_{3}+A_{4})\\&+det(A_{1})+det(A_{2})+det(A_{3})+det(A_{4})\end{aligned}}$

Again it is worth pointing out, that the formula works only for large sums of matrices. If we try to use it to calculate the determinant of just two matrices, we'll get an error:

$\left|{\begin{array}{cc}a+\alpha &b+\beta \\d+\delta &e+\epsilon \end{array}}\right|-\left|{\begin{array}{cc}a&b\\d&e\end{array}}\right|-\left|{\begin{array}{cc}\alpha &\beta \\\delta &\epsilon \end{array}}\right|=e\alpha +a\epsilon -d\beta -b\delta$

Removed tag at top

I have removed the tag at the top after coming here from WT:V and looking at what the tag was about. There seems to be no major issue, just a mix up of how something is represented sometimes with what it is. Dmcq (talk) 08:37, 14 May 2011 (UTC)

Discussions elsewhere

There have been extensive discussions regarding this article many places elsewhere including my talk page, Michael's talk page and wp:ver. . IMHO Michael P. Barnett exhibits so much caution and politeness and interprets feedback in that same manner that he read it that he had been shut down here. I think he's ready to come back, if so please understand this situation. North8000 (talk) 11:19, 19 May 2011 (UTC)

Goodness. I had no idea the exchanges above had led to such a firestorm. We could have had a more productive time with more light and less heat had Michael kept the discussion here, since this is the article he wants to improve. I agree whole-heartedly with your remarks at WT:V, and I hope Michael doesn't feel that we are trying to squelch discussion or edits.

My impression is that the main difficulty with the lede is that it does not clearly distinguish between the determinant of a matrix and the determinant operator det:S^n×n → S. I think it would be a mistake to try to give a rigorous definition of the det operator in the lede. Something like "For a given commutative ring S and non-negative integer n the determinant operator det:(Sⁿ)ⁿ → S is the essentially unique n-ary skew-symmetric operator with the following properties ... " would scare everyone away instantly. It makes more sense to start with the determinant of a matrix, and lead into the determinant operator mapping matrices to numbers. Something like:

In algebra, the determinant of a square matrix with entries from a number structure S (a field like the real numbers, or more generally a commutative ring) is a number from the same underlying structure S. The determinant operator det that maps n×n matrices over S into values from S does so in a manner that preserves multiplication — det(AB) = det(A)·det(B) — and such that a matrix A is invertible over S if and only if det(A) is an invertible member of S.

Already this is arguably too general. It's hard to introduce det with suitable generality right off the bat, and yet common examples like the Jacobean are already technically determinants over rings, not fields. Perhaps it makes sense to start even more specifically: "The determinant of a square matrix over the real numbers is a real number, assigned in such a way that det(AB) = det(A)·det(B), and such that A is invertible if and only if det(A) is nonzero. More generally, determinants can be defined for matrices over other fields, like the complex numbers, or even over an arbitrary commutative ring (where A is invertible if and only if det(A) is an invertible member of the ring)." After that it would make sense to observe that the determinant operator can be seen as an n-ary skew-symmetric operator etc. etc. The main article needs a section early on defining the determinant operator this way.

I agree with Michael that the geometric interpretation doesn't belong in the next breath of the lede, but I do believe it deserves an example in a subsequent paragraph somewhere in the lede.

-- Elphion (talk) 18:12, 19 May 2011 (UTC)

Thanks. My main thought is that you are dealing with someone who is unusually cautious and polite and to please keep that in mind. Sincerely, North8000 (talk) 20:04, 19 May 2011 (UTC)

Restatement of concerns based on further reading

As a courtesy to the editors who encouraged me to follow up my previous efforts regarding this article, I spent some time yesterday consulting further sources. I am restating my concerns here, taking this further material into account, but I do not plan any further action. If anyone else thinks any of my concerns have merit, they can act. Otherwise I accept that I have misunderstood the role of Wikipedia.

As a non-mathematician who does know a little mathematics, particularly what is taught to science students, I think the lead takes a far more abstract and esoteric approach to the subject than has been customary in authoritative accounts from the time determinants were invented to the present, and uses mathematics outside the material typically taught to science students, let alone what is known to the general public. The standard approach can be understood by anyone who can add and multiply. I have questioned and still question the verifiability of some statements in the lead. Also, I question further the neutrality of "the geometric aspect ... is the key to understanding ... and is not adequately represented in the current article" which already presents this idea disproportionately in comparison with sources I cite below. Several statements in the lead are misleading.

Books that I have consulted include those listed in the following collapsed paragraph.

Extended content

Kreyszig's text The textbook Advanced Engineering Mathematics by Erwin O. Kreyszig in collaboration with Herbert Kreyszig and Edward J. Normington, John Wiley & Sons, Inc. 10th edition, 2011 [2] contains extensive coverage of determinants. This book is being shipped in large volume to university bookstores at present. The WK article about Kreyszig begins: Erwin O. Kreyszig ... (1922 ... 2008) was ... a pioneer in the field of applied mathematics: non-wave replicating linear systems. He was also a distinguished author, having written the textbook Advanced Engineering Mathematics, the leading textbook for civil, mechanical, electrical, and chemical engineering undergraduate engineering mathematics. The "Purpose and Structure of the Book" (p. vii) states: "This book provides a comprehensive, thorough and up-to-date treatment of engineering mathematics. It is intended to introduce students of engineering, physics, mathematics, computer science and related fields to those areas of applied mathematics that are most relevant for solving practical problems. A course in elementary calculus is the sole prerequisite. ... (the book adopts) a modern approach ... " This book contains a substantial section on Determinants. Geometrical interpretations are introduced as a team project in the penultimate exercise, that takes up less than 1% of the entire section. It is couched in terms of points, lines, squares, circles and spheres, with their common usage, rather than measure theory. The texts by Bretscher, by Strang and by Artin Otto Bretscher, Linear Algebra with Applications, 4th ed. Prentice-Hall, 2008. On p. ix, under Continuing Text Features, the author states "Linear transformations are introduced early ... Visualization and geometrical interpretation are emphasized extensively throughout". Gilbert Strang, Linear Algebra and its Applications. The author makes similar remarks in front matter. His first diagram is on p. 3. It is essential to note, however, that the first geometrical topics mentioned in these texts are utterly different from the geometrical topic in the lead. These examples relate to the solution of linear systems as the intersection of lines and planes. Bretscher introduces a simple case of the idea that comprises the second sentence of the lead, but not until p. 85, where he proves it as a formal Theorem -- not a definition, not an explanation and without comment on how it helps understanding. Michael Artin, Algebra. This DOES give the transformation of a unit square on p. 18. However, this does NOT try to encompass multidimensionality, which the present lead sentence does. And Artin writes from the specialized viewpoint of an algebraic geometer. Encylopedic works In my previous posting I mentioned the consistent treatment of determinants in the pre-1990 works in the list that follows. These CANNOT be ignored. These (in particular Encyclopedia Britannica), are amongst the reference works in a vast number of public, university and school libraries. They express the knowledge learned by older readers who seek refreshment of their recollections. The Oxford University Press and the Cambridge University Press mathematics encyclopedias published in 2003 and 2011 treat determinants traditionally, and the introductions in each can be understood by anyone who can add and multiply. The CUP compendium mentions the 3-dimensional geometrical interpretation, using vector notation (not measure theory) in a short, final paragraph. 1. Encyclopedia Brittanica 2. Encyclopedic Dictionary of Mathematics of the Mathematical Society of Japan, English translation, published by MIT press,that has forewards by the President of that society and the President of the American Mathematical Society 3. Compendium of Applicable Mathematics, edited by Karel Rektorys 4. Eberhard Zeidler, Oxford User's Guide to Mathematics, Oxford University Press, 2003. 5. K.F.Riley and M.P.Hobson, Essential Mathematical Methods for the Physical Sciences, Cambridge University Press, 2011.

Because editors who are not familiar with the topic may be following this discussion, I think the following collapsed explanation may be helpful.

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Some simple examples I begin with some simple examples. Starting in this way is consistent with the way many published accounts of determinants have started, since determinants were invented in the 1700s to major texts and encyclopedias published in 2011. The following tabulation of four numbers, enclosed within a pair of vertical lines, is called a determinant. The number of rows equals the number of columns. This number is called the order of the determinant. The individual items are called the elements of the determinant. ${\begin{vmatrix}1&2\\3&4\end{vmatrix}}$ This is a shorthand for 1 × 4 - 2 × 3 = 4-6 = -2. Correspondingly, ${\begin{vmatrix}x&y\\u&v\end{vmatrix}}=x\times v-y\times u$ The determinant of order 3, that consists of the elements 1, 2, …, 9, is written ${\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}}$ This is a shorthand for $1\times {\begin{vmatrix}5&6\\8&9\end{vmatrix}}-2\times {\begin{vmatrix}4&6\\7&9\end{vmatrix}}+3\times {\begin{vmatrix}4&5\\7&8\end{vmatrix}}=$ $1\times (5\times 9-6\times 8)-2\times (4\times 9-6\times 7)+3\times (4\times 8-5\times 7)=$ $1\times (45-48)-2\times (36-42)+3\times (32-35)=1\times (-3)-2\times (-6)+3\times (-3)=-3+12-9=0$ This follows the following prescription: 1. Take the first element in the first row, in this case 1. 2. Strike out the row and column that contain the first element in the determinant that we started with. This leaves the determinant (of order 2) ${\begin{vmatrix}5&6\\8&9\end{vmatrix}}$. 3. Multiply the results of steps 1 and 2. 4. Take the actions corresponding to steps 1 to 3, using the second element in the first row. This gives $2\times {\begin{vmatrix}4&6\\7&9\end{vmatrix}}$. 5. Subtract the result of step 4 from the result of step 3. 6. Take the actions corresponding to steps 1 to 3, using the third element in the first row. This gives $3\times {\begin{vmatrix}4&5\\7&8\end{vmatrix}}$. 7. Add the result of step 6 to the result of step 5. This fits the following general pattern that holds for determinants of any order. Work through the first row, element by element. For each of these elements in turn: 1. Multiply it by the result of striking out the row and column in which it is located. (In this prescription, the row is always the first). 2. Subtract the second of these results from the first. 3. Add the third. 4. Subtract the next, if the order is more than 3. 5. Continue in this way, alternately adding and subtracting, until the end of the first row has been reached. This prescription converts a determinant of order 100 into a sum that contains 100 determinants of order 99. The prescription turns each of these into a sum that contains 99 determinants of order 98. And so on. If the determinant contains a lot of zeroes, the situation is not so bad, but allowance has to be made for when this is not the case. There are simple notations for writing the prescription concisely, but these do not need to be explained yet. A question my students in a graduate course on computer literacy for humanists would have asked at this point was "why bother us with this -- why are determinants useful?" My answer would have fallen back on the problem: "2 oranges and 3 apples cost 60 cents. 3 oranges and 2 apples cost 65 cents. Can we calculate the cost of an orange and the cost of an apple from this information?" The question is not "what are the prices", but "can we determine them". The answer is yes. Next question "how do you know that the answer is 'yes'". Answer: "because the determinant ${\begin{vmatrix}2&3\\3&2\end{vmatrix}}$ is not zero. Suppose the information was 2 oranges and 3 apples cost 60 cents. 4 oranges and 6 apples cost $1.20. Then we would not have enough information. I can tell this because the determinant ${\begin{vmatrix}2&3\\4&6\end{vmatrix}}$ is zero." Next question: "But suppose you were given 20 shopping receipts for different amounts of the same 20 kinds of food, it would take ages to compute the determinant by the process you described." Answer: "There are several theorems that enable the conversion of one determinant to another that has the same value and takes less time to compute because it contains more zeroes." I did not actually take this route in class. The module on matrices focused on linear transformations, using word problems and worked solutions that consisted of simple sentences to explain: 1. The respective costs of an apple and an orange can be found "graphically" (by plotting on squared paper) and observing where the straight lines with the formulas 2x+3y=60 and 3x+2y=65 intersect. 2. The meaning of "linear expression" (such as 2x+3y), "linear equation" (such as 2x+3y=60), "linear transformation" (such as: convert the number of jam tarts and muffins to be shipped into the pounds of sugar and flour these require) and "linear system" (lists of linear equations that can be solved collectively), that all have names that follow from the fact that they deal with expressions that give straight lines when plotted. 3. The definition and basic properties of matrices. 4. The multiplication of matrices to express compositions of transformations (for example, to find the number of pounds of flour and sugar needed to supply Dainty Teas and Hearty Arty's with jam tarts and muffins, using the numbers of each that they require, respectively). (I mention this for readers who seek clarification of "linear" that pervades Determinants and related topics. I am not trying to fork or sneak in unpublished work. It has been reported.)

The lead, sentence by sentence

• 1. "characteristic number" has a specialized technical meanings in algebra that is irrelevant to this article.
• 2. "volume" is being used here with a specialized meaning that the article links, for an explanation, to measure theory which is outside the content of the Kreyszig book (see collapsed data above) and many other math texts for non-mathematicians.
• 3. For whom is this article intended?
• 4. "straight forward" -- to whom?
• 5. The feedback to my request for a reference supporting the second sentence claims it is verified clearly in the body of the article. WHERE???
• 6. "are important both in calculus where they appear in the Jacobian ... and in multilinear algebra". This suggests that besides an incomprehensible (to the common science graduate) use that needs knowledge of measure theory to understand, the two other uses are for Jacobian in calculus (ignoring other uses in calculus) and in multilinear algebra, ignoring the VAST utilization in linear algebra that is the basis of its central role in scientific and engineering computation.
• 7. The lead now goes on to use further terminology that is completely unnecessary and off-putting to readers who want the kind of information in any of the sources I cite in the collapsed data above (except Artin's book).

The societal need to make mathematics intelligible has a considerable literature, that includes reports of the National Academy of Sciences, educational agencies and educational literature. Need WK be responsive? Michael P. Barnett (talk) 02:38, 21 May 2011 (UTC)

The meaning of characteristic is the typical meaning in a dictionary

"1. Also, char·ac·ter·is·ti·cal. pertaining to, constituting, or indicating the character or peculiar quality of a person or thing; typical; distinctive: Red and gold are the characteristic colors of autumn."

You are welcome to try and say it better but there is no jargon there.

I agree having measure there is overkill. I can see a problem with just saying volume as it is area for 2 dimensional matrices and hypervolumes for n-dimensional one and just scale for a scalar. The standard term is just volume for the lot but it might be better to expand a bit here.

The first few bits of articles should be pitched at a level where people with a grasp of most of the basics for an elementary introduction can read it to get the basics. The lead may contain some more advanced material at the end as it has to also summarize the article.

Straightforward to practically anyone who will be able to grasp the basics..

By practically any book on it in the references I'd have thought, do you actually really doubt this or what's your point? Why do you think they occur in the denominator when inverting matrices or why there are all those pictures of skewed parallelograms or parallelepipeds about?

The bit about the Jacobian is part of the statement of notability for the topic, all topics should say why they are notable in the lead. As far as I'm aware the uses as stated are the major reasons why determinants are useful besides just being used as an intermediary in inverting matrices which is hardly a reason for major notability. There's lots of other important determinants but the Jacobian is by far the most important in engineering and science.

I agree the second paragraph should be moved to the end of the lead as it is a summary of more advanced uses and should be marked as more advanced.

Are you sure you're not conflating the large treatment of matrices in books with the treatment of determinants? Dmcq (talk) 12:16, 21 May 2011 (UTC)

OK I volunteer to be the dummy. I'm an engineer, my work is in engineering / scientific fields, I devour technical writings. I learned a little about matrices and determinants in school and forgot it all and never used it. I just read the lead and learned absolutely zero / got no clue from it regarding what a determinant is. IMHO it is just a jargon-loaded dance around the edges of it without really defining what it is. In order to not jeopardize my dummy status, I'll avoid reading and learning the body of the article.  :-) North8000 (talk) 12:34, 21 May 2011 (UTC)

[Reply to initial post, pushed down below other replies due to edit conflict] Dear Michael P. Barnett, please allow me to suggest sincerely that you are putting too much effort into this issue. I sympathise, but so much text on a talk page makes it virtually impossible to reply (it also risks errors, as in you computation of a 3×3 determinant that by inspection should have given 0, but which WP policy does not allow me to correct). I'll try to reply just to the point-by-point list

1. "characteristic" is of course a silly word. It should go.
2. "volume" is being used with its everyday meaning, assuming the space is of dimension 3. The link to measure theory could be useful (depending on the quality of that article) for those wondering how volume can be defined in general, but this is not prerequisite to understanding this sentence. I already commented earlier on unfortunate aspects of the "measure of volume" phrase, and would welcome improvement. If determinants are around since the 1700s (which was news to me, and makes them about twice as old as matrices), concern about area and volume have been around since the earliest signs of civilization (of course linear transformations are not quite so old).
3. I don't really understand which part of the lead is targeted by this question. The answer is of course anybody interested in determinants and their use, including but not limited to students of science and engineering.
4. "straight-forward" is a kind of apologetic term that usually contributes little. This case is no exception. But calling "this transformation multiplies area by 2" a straight-forward statement does not shock me.
5. I don't recall any discussion about verification of the second sentence. Do you doubt that it is correct? The body of the article does discuss this point under "Applications", as I think I mentioned before. That is about as late in the article as the cited sentence is early, very curious. But that was not true the first time I mentioned this, a month ago or so.
6. The question suggests that understanding the fact that area or volume are multiplied by some factor is incomprehensible to the common science graduate. This would not be my assessment of their mental capabilities. Why are you so obstinate about this statement; my experience is that it causes absolutely no difficulty. Saying this is incomprehensible suggests that you take an extreme formalistic point of view (yes defining volume properly in a very general sense is hard, but it is very intuitive notion that directly inspired differential calculus, any only led to measure theory much later). The multilinear algebra reference is weird, I never really understood what is meant by that. From my personal experience, the major reasons for introducing deetrminants in the undergraduate math curriculum appears to be twofold (I think the sentence attempts to address those two points, in the opposite order):
   1. The are needed to define characteristic polynomials
   2. They are needed for doing change of variables in multivariate integration
7. Is what you find off-putting the talk about fields and commutative rings? This does address a somewhat more mathematically mature audience, but does touch one the essence of determinants (unlike the volume interpretation, IMO). One could make this more broad-public by saying that a linear system of equations with as many equations as unknowns is uniquely solvable if and only it its determinant is nonzero. Personally I feel however that viewing determinants as expressions where the coefficients are assumed to lie in a field goes somewhat against their nature; the most fundamental properties of determinants are related to the fact that they only require a commutative ring (i.e., they do not involve any divisions). The definition of the characteristic polynomial is a good example of this; there is absolutely no need to consider rational functions (nor interpreting polynomials as polynomial functions) to understand that definition. I'm not sure how this should be reflected in the lead however.

I'd like to have some other opinions about this though, before tinkering with the lead. Marc van Leeuwen (talk) 12:41, 21 May 2011 (UTC)

I've put in a couple of examples to cut down the impact of field and ring and removed the reference to measure theory. The uniqueness seemed unnecessary in the lead and I moved the symbolism bit a bit up.

There's a bit of wordsmithing that cold be done on things like characteristic, I guess if you know a lot of maths you might think it had a formal meaning here so yes it should go but what should it be replaced with? The bit about straightforward could go too without any loss as it is explained immediately but I haven't figured out how to rephrase that bit either. Dmcq (talk) 13:04, 21 May 2011 (UTC)

I've made some further changes, including some trimming-down, but also mentioning the relation to the solvability of linear systems, which I think is the prime motivation for determinants in linear algebra. I also included a reference to Cramer's rule, which is maybe not that crucial (and is likely to provoke some hostility from the crowd convinced that Cramer's is a bad bad rule, and by contagion determinants are bad), but which I felt was useful to counter the obvious "what's the relevance of determinants if all you care about is whether they are zero or not". Marc van Leeuwen (talk) 14:29, 21 May 2011 (UTC)

The last paragraph already talks about inverting matrices so there's duplication there. Plus I think you've put in rather a lot of padding words which don't really help much. You don't need determinants to solve linear equations and they just appear in passing is my belief if you solve them using for instance gaussian elimination. Dmcq (talk) 14:43, 21 May 2011 (UTC)

Very grateful to respondents. IMHO article is vastly improved. Have non-contentious (I hope) information about infinite determinants (in series solution of Mathieu equation re lunar motion, Schrödinger equation for two-electron atoms -- and I do mean determinants as well matrices), symbolic calculation of determinants (so-called Markov algorithm actually published earlier by a physical chemist) etc. Sorry about 4 times 8 = 48. Hope you believe it was just exhaustion. More anon, with less verbosity. Michael P. Barnett (talk) 14:59, 21 May 2011 (UTC)

I don't see that anything about atoms or Markov algorithms would be suitable for the article. They just are not really relevant to the topic, they're just some algorithms or uses. There's a mention of infinite determinants at the end. Dmcq (talk) 15:33, 21 May 2011 (UTC)

Suggestions for added content

This replies to Dmcq's most recent comment, but I thought it better to start a new section.

Symbolic calculation of determinants

Are we referring to the same Markov algorithm? I mean the algorithm that is used in symbolic calculation software to construct symbolic expressions of symbolic determinants. Consider a determinant of order n, that contains elements a + b x, where a and b stand for real numbers and x stand for a symbol. Expansion gives a polynomial of degree n in x. Compute the numerical values of the determinant with x replaced by 0, 1, ..., n-1. Take differences. Hence the coefficients in the characteristic polynomial. This is described in K.O.Geddes, S.R.Czapor and G.Labahn, Algorithms for computer algebra, Kluwer, 1992. By extension, I worked with determinants containing elements a + b x + c Z. I needed to push the order of the determinant as far as possible in a list that included 308. This was the highest (in the list) that the system could handle when each element contained at most one variable. So my co-workers constructed the characteristic polynomials in x for Z=0,1,2,... and we took differences, to construct a characteristic polynomial in x, in which coefficients were polynomials in Z (as an explicit symbol). Faddeyev and Faddeyeva refer to this as the Markov algorithm, in Computational Methods of Linear Algebra, tr. R. C. Williams, Freeman, 1963. In fact, it was invented prior to Markov's formulation by B. L. Hicks, Journal of Chemical Physics, 8, 569, 1940. Hicks work was described in the standard text on theory of infra-red spectra by Wilson, Decius and Cross and it was the preferred method for constructing characteristic polynomials in theoretical chemistry for about 20 years. In suggesting mention of the general method I did not imply referencing how I used it.

Infinite determinants and differential and algebraic equations

Are we talking about the same infinite determinants? I learned about these during a brief encounter with the Mathieu equation about 60 years ago that led me to Hill's work on lunar theory and to Section 2.8 of Whittaker and Watson. Active research and publication continues on these interrelated topics, e.g. Curtis Wilson, The Hill-Brown Theory of the Moon's Motion, Springer, 2010; L. Wille and R. Phariseau, On the computation of a certain class of Hill determinants, Journal of Computational and Applied Mathematics, 15, (1) 83-91, 1986; I can run SCOPUS and Web Of Science searches on this if need be. Whittaker and Watson mention Fürstenau's work on algebraic equations, and the continuants of Sylvester. The "hand waving" explanation, that I have used in seminars for how infinite determinants turn up in calculations of planetary theory and quantum mechanics is by extension of the Frobenius method that expands a differential equation in a series of special functions (e.g. Laguerre polynomials, and establishes a recurrence relation between successive coefficients. In this approach to the "helium problem", (one of the most famous achievements of Chaim Pekeris), this contains 33 terms, that overlap from equation to equation, giving an infinite set of simultaneous equations in the coefficients. These contain the variables I labeled x in the preceding paragraph.

Determinants and bilinear forms

I remember looking at Muir's monograph about 20 years ago, apropos the "helium problem", and finding material on bilinear forms that I experimented with, but cannot recall why. So I just ran a quick search for a connection and found a paper that looks quite mathematical: Xuanting Cai, A Gram Determinant for Lickorish's Bilinear Form, [3] indexed under Mathematics>Geometric Topology.

Resultants and polynomial systems

Recent work on Dixon resultants, e.g. Robert Lewis, Heuristics to accelerate the Dixon resultant, Matcom 77 (4) April, 2008 [4] is an important alternative to the Gröbner basis approach to the solution of polynomial systems of major importance in theoretical studies of protein folding and other practical problems (there is an overview of these in Manfred Minimair and Michael P. Barnett, Solving polynomial equations for chemical problems using Gröbner bases, Molecular Physics, 102, (23-24) 2521-2535, 2004.) Might there be some mention and/or further links to some of these topics/people and to bilinear forms and elimination theory and suchlike?

My non-neutral point of view

The chance of anyone who is still following this discussion looking at my joint paper on Gröbner bases is "vanishingly small" (I really am trying to write wiki-ese) but I had better come clean now. The whole point of the paper was to explain the operation of Buchberger's algorithm without falling into the usual style, so eloquently described by the Wigner medalist Harry Lipkin: "As a graduate student in experimental physics ... all attempts to follow a lecture (on certain algebraic topics) resulted in a losing battle with characters, cosets, classes, invariant subgroups, normal divisors and assorted lemmas. It was impossible to learn all the definitions used in one lecture and remember them until the next. The result was complete chaos. It was a great surprise to learn later on the (1) the techniques can be useful, (2) they can be learned without memorizing the large number of definitions and lemmas which frighten the uninitiated." Michael P. Barnett (talk) 19:59, 21 May 2011 (UTC)

No nothing there sounds in the least suitable for this article. They are not about determinants. You should set up separate articles. Dmcq (talk) 20:18, 21 May 2011 (UTC)

I have had a read of what you wrote above and much of it makes no sense to me even though I am familiar with most of the individual concepts like Markov algorithms and the Mathieu functions and Gröbner bases. I am not a professional mathematician so it is quite possible it does make sense but I am pretty certain that if it does it is at a level which has not been reached by most of Wikipedia yet and they haven't made general notability yet. So I don't think they would even be suitable for separate articles. Dmcq (talk) 20:41, 21 May 2011 (UTC)

Sorry. I was trying to be constructive in response to how the article has improved, got carried away with enthusiasm, and it misfired. The less I write now the better. But I would like to remedy one point asap. I should have said "an algorithm attributed to Markov that is completely unrelated to the famous Markov algorithm". The authors of what was, for some time, the standard Russian text on linear algebra use the term "Markov algorithm" for the process I mentioned. Maybe this shows their parochialism (or maybe it predated). I was trying to be terse. Your reaction has been very salutary. I have fallen into the trap of assuming that specialized pieces of information known to me are known to everyone else. Sixty years ago the construction of numerical tables from "differences" was all pervasive in computation. After I posted my message I noticed the "to-do" list. This includes algorithms to compute symbolic determinants! What I tried, but failed, to describe in a simple fashion, is a standard method, which has accessible verifiability. May I have another crack at it in a few days time. Incidentally, I assumed that one of your specializations is abstract algebra. Michael P. Barnett (talk) 21:14, 21 May 2011 (UTC)