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

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, matrices with multidimensional indices

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]

Article wrong by many omissions?

The Example about "In a triangular matrix, the product of the main diagonal is the determinant"

There are many triangular matrices which can be derived. Unless you state more constraints (e.g. Hermite Normal Form or alike), you cannot simply postulate that the diagonal product is the determinant. For example, if I produce some (remember - it is not unique) upper triangular matrix with row operations, instead of column operations, I cannot reproduce the findings in the example. So, either constraints or insights are clearly missing.

2003:E5:2709:8B91:29A5:FE96:3F33:E513 (talk) 07:50, 4 August 2022 (UTC)

No, in any upper or lower triangular matrix, the determinant is the product of the main diagonal: all the terms in the expansion of the determinant along any column are 0 except for the term involving the cofactor determined by the matrix entry at the intersection of the column and the main diagonal -- and by induction the determinant of that cofactor is the product of its main diagonal (which is the rest of the diagonal of the full matrix). The explanation following the example tacitly assumes that the matrix is nonsingular, but the claim remains true for singular triangular matrices as well. -- Elphion (talk) 14:05, 4 August 2022 (UTC)

"the linear map" in the introduction

There are two mentions of "the linear map" associated to a matrix and one to "(the matrix of) a linear transformation". The word "the" in all of them is incorrect, since in each case noun referred to is not unique.

The first two occurrences should be replaced with "a". The last sentence is correct without the parenthesized clause, as a sentence about the determinant of a linear transformation. Alternatively the parenthesis can be adjusted to "a matrix representing". Thatwhichislearnt (talk) 14:32, 28 February 2024 (UTC)

"The" is correct, since there is exactly one matrix that represents a given linear map or transformation on given bases. When one use "the matrix of a linear map" this supposes that bases are implicitely chosen. D.Lazard (talk) 15:10, 28 February 2024 (UTC)

None of the sentences mention the basis. Thus it is incorrect. Thatwhichislearnt (talk) 15:17, 28 February 2024 (UTC)

For example, there is the sentence "Its value characterizes some properties of the matrix and the linear map represented by the matrix.". There is no such thing as "the linear map represented by the matrix". Thatwhichislearnt (talk) Thatwhichislearnt (talk) 15:19, 28 February 2024 (UTC)

Also, that "convention" made up by you that "this supposes that bases are implicitely [sic] chosen" need not be the assumption of a general reader of Wikipedia, even more of someone that is just beginning to learn about determinants and linear transformations. It is precisely a common point of failure that students misinterpret the correspondence between matrices and linear maps as one-to-one. Thus explicit is better than implicit. Thatwhichislearnt (talk) 15:31, 28 February 2024 (UTC)

As far as I know, for most readers, bases are always given, and they do not distinguish between a vector and its coordinate vector. So, it is convenient to not complicate the lead by discussing the choice of bases. However things must be clarified in the body of the article, although there are other inaccuracies that are more urgent to fix. D.Lazard (talk) 15:54, 28 February 2024 (UTC)

Well "they do not distinguish between a vector and its coordinate vector" and "bases are always given" are both fundamental errors. And regarding "for most readers", citation needed, if anything those would be the readers that have not learned the content properly or haven't learned it yet. Thatwhichislearnt (talk) 16:02, 28 February 2024 (UTC)

Also, it is not a complication using "a" instead of "the". For a reader without sufficient attention to detail the wording might not be noticed. Yet, the article wouldn't be lying to them. On further readings they might notice. I agree that between the two choices that don't lie to the reader: Using "a" and using "the matrix + mentioning the basis" using "a" is the one that introduces no complication to the articles' introduction. Thatwhichislearnt (talk) 16:08, 28 February 2024 (UTC)

Reverted your edit to give you the opportunity to fix the absurd edit message. Also here you complain about " to not complicate" and then you choose the option that is the more complicated? Thatwhichislearnt (talk) 16:23, 28 February 2024 (UTC)

I maintain that the indefinite article is wrong here. If you disagree wait a third person opinion. In any case, do not edit war for trying to impose your opinion. D.Lazard (talk) 17:30, 28 February 2024 (UTC)

D.Lazard's phrasing is much better, and more informative. -- Elphion (talk) 17:57, 28 February 2024 (UTC)

You maintain? On what basis? Where is the citation? Plus it also your opinion that the introduction should not be complicated. Thatwhichislearnt (talk) 18:00, 28 February 2024 (UTC)

For citation, any linear algebra text. -- Elphion (talk) 18:03, 28 February 2024 (UTC)

No no. That is not what I am asking. Citation for using "a" being wrong. Again all of those linear algebra texts say that the matrix does depend on the basis. Thus a lack of mention of the basis yields the article "a". Thatwhichislearnt (talk) 18:05, 28 February 2024 (UTC)

That is not his phrasing. That is one of the phrasings that I said should be done and he objected ("to not complicate"). Thatwhichislearnt (talk) 18:03, 28 February 2024 (UTC)

It is always the same issue with him. Editing Wikipedia became his entertainment in retirement and all over the place defends incorrect wording on the basis of "simplicity". Thatwhichislearnt (talk) 18:09, 28 February 2024 (UTC)

Sorry, I meant the text resulting from D.Lazard's edit of 17:24. I don't care whose text it is, it is superior to using just an indefinite article. The key point of matrices is that for a given choice of bases there is a 1-1 correspondence between linear transformations and matrices of appropriate size. That's where the definite article comes from. And please refrain from attacking another user; keep the discussion on the article. -- Elphion (talk) 18:16, 28 February 2024 (UTC)

The entire reason why I posted this section. The initial version of the article was wrong for implying there is "the matrix of a linear map". Then his "opinion" passes through the following stages:

1. Gaslighting that there is some made up convention that bases are implicitly assumed. That could work with non-mathematicians, but there is no such thing.

2. That do not complicate the article.

3. The (demonstrably false) opinion that "a" is wrong. When clearly when one does not make a choice of basis the association between linear maps and matrices is a one to many one. Thatwhichislearnt (talk) 18:46, 28 February 2024 (UTC)

Also, note that his edit, that you consider superior, was still leaving another occurrence of the same mistake, the mention in the part about the orientation. Thatwhichislearnt (talk) 18:50, 28 February 2024 (UTC)

And the grammar was also inadequate. Thatwhichislearnt (talk) 18:52, 28 February 2024 (UTC)

Look, I agree with D.Lazard that the definite article is superior; I agree with you that some reference to choice of bases is appropriate. And I repeat, casting shade on a fellow editor ("gaslighting" above) is not helpful, and will eventually get you blocked. -- Elphion (talk) 19:06, 28 February 2024 (UTC)

The helpful way to proceed at this point is to suggest a concrete prospective wording here on the talk page so we can discuss it. -- Elphion (talk) 19:20, 28 February 2024 (UTC)

All the occurrences are fixed now. For the last one that was left unfixed, regarding orientation, I used "a" again. In that case it is talking about determining orientation. For orientation every matrix of the endomorphism, in every base, can be used. Do you also think "the" + "basis" is better in that case? The "a" removes the error of "the" + no"basis" and it allows for the whole picture of independence of the basis. Thatwhichislearnt (talk) 19:29, 28 February 2024 (UTC)