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]

Using column vectors to represent points

I believe that there was a time when geometric points were often represented by row vectors, but now they are usually represented by column vectors. I do not have any evidence for the first part of that statement, but for the second part I have found:

• [2] written in 1993 which says "Recent mathematical treatments of linear algebra and related fields invariably treat vectors as columns".
• [3] which says "The general convention seems to be that the coordinates are listed in the format known as a column vector".
• Olver and Shakiban (Applied Linear Algebra, 2018) who say that the term "vector" without qualification means a column vector.
• [4] where a comment says "we typically write the coordinates of our points as columns".
• The article transformation matrix which uses column vectors.

If most people learn that points are represented by column vectors, then the 2D example at Determinant#Geometric meaning would be easier to understand if it just used the columns of A. It would talk about the vertices at (0, 0), (a, c), (a + b, c + d), and (b, d).

It would need a new image instead of File:Area parallellogram as determinant.svg. Also I think the proof about the signed area would need to use v^⊥ instead of u^⊥, although I have not completely worked that out. The section could still mention that the determinant of the transpose gives the same result.

Also, please note that the 3D example in that section already uses the columns of A. JonH (talk) 01:20, 10 July 2024 (UTC)

You must distinguish between vectors of $\mathbb {R} ^{n}$ with are tuples and commonly denoted in a row between parentheses such as $(a_{1},\ldots ,a_{n}),$ and the corresponding row and column vectors that are matrices and are denoted between square brackets. In other words, a vector is a n-tuple that can be represented with either a $n\times 1$ matrix (column vector) or a $1\times n$ matrix (row vector). You are true when saying that the common convention for matrix computation is to represent vectors with their associated column matrix.

I did not find anything in the linked section that goes against these common conventions. However, the wording is rather confusing, and could certainly be improved. D.Lazard (talk) 09:02, 10 July 2024 (UTC)

At a second thought, the main confusion of this paragraph is that it confuse points, the tuples of their coordinates and the corresponding row and columns vectors. D.Lazard (talk) 09:12, 10 July 2024 (UTC)

Tuples and row vectors (or column vectors, depending on the source) are so commonly conflated in both use and notation that any pedantic clarification here needs to be written very carefully. Notation here is also far from standardized (tuples can be written with square, round, or angle brackets; matrices can be written with square or round brackets). Also points in Euclidean space (or geometric vectors in a Euclidean vector space) are not tuples per se, but can be represented as tuples relative to an arbitrary Cartesian coordinate system. The object and its representation as numerical data are also commonly conflated. –jacobolus (t) 13:04, 10 July 2024 (UTC)

Include other notions of determinant (of endomorphism and family of vectors)

Hello @D.Lazard and others. I would like to at least partially restore my last edit https://wiki.riteme.site/w/index.php?title=Determinant&diff=1248975782&oldid=1248503608, for the following reasons :

• I think it is very wrong to treat the determinant nearly only with respect to matrices, as the article does. The two other types of determinants (of an endomorphism or of a family of vectors, which I forgot to add) are very fundamental and omnipresent concepts and deserve more than a mention in passing. All three concepts should be fully introduced in the lead. The determinant of a linear map is a core concept in physics (e.g. continuum mechanics), while the determinant of a family of vectors is important for instance in relation to the orientation of a basis or the triple product in an Euclidean space. For comparison, the French and Spanish language articles do not treat the determinant as only a matrix concept in the lead.
• I think the alternative definitions of the determinant of an endomorphism I gave in the Definition section were correct and added value to the article. (I forgot to provide sources; there is for instance https://www.worldscientific.com/doi/pdf/10.1142/9789813142688_0001?srsltid=AfmBOoq_4llX_GZvzh9yp1x4hUPgJg--RFi8mSXk37naAz6hF55a9_wp on page 20)
• I tried to abide by WP:TECHNICAL and my edits were as clear and detailed as possible.
• As for now, the article sometimes lacks logical structure. The determinant of an endomorphism shouldn't belong in the "Abstract algebraic concepts" section but rather in "Definition"; also, "Characterizations of the determinant" would make more sense in "Definition" than in "Properties of the determinant" (I didn't change that yet); moreover, the special cases of 2x2 and 3x3 matrixes should be introduces after the general case rather than before it.
• I think it would be better if other contributors improved specific points in the lead rather than reverting the whole of my edits.

Uttercrap80 (talk) 15:17, 2 October 2024 (UTC)

I agree that the structure of the article requires improvements, but your edit goes against the Manual of Style: The lead should stand on its own as a concise overview of the article's topic.

In particular:

• The determinant of an endomorphism is mentioned in the lead, and a section is devoted to it. More details do not belong to the lead and must appear in the relevant section. The determinants of a family of vectors and of a linear map are defined only when bases are fixed. So, they are not well defined concepts; However, they may be mentioned among the applications, although these determinants are simply the determinants of the matrices that represent them and are defined only if this is a square matrix.
• On the opposite, the definition of the determinant in term of its properties does belong to the lead and must be moved into a subsection of § Definition.

Also

• The heading § Abstract algebraic aspects must be removed and its subsections must be moved into subsections of § Generalizations and related notions. The right place of this section may be discussed.

D.Lazard (talk) 13:48, 3 October 2024 (UTC)