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| Somebody (myself, if I'll win the laziness) should add something about the formal definition of determinant (an alternating function of the rows or columns etc. ...), of which its unicity and how to compute it are consequences. --] | |||
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| == Right handed coordinante == | |||
| ...and add to that the foundation of the definition, which is something to do with multilinear functions.<br> | |||
| Also worth mentioning that historically, the concept of determinant came ''before'' the matrix. | |||
| the following sentence is not clear. | |||
| :That would certainly be very interesting. What is the history of the concept? --AxelBoldt | |||
| "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 == | |||
| I'll see what I can dig up, but briefly: a determinant was originally a property of a system of equations. When the idea of putting co-efficients into a grid came up, the term "matrix" was coined to mean "mother of the determinant", as in womb. | |||
| <br> | |||
| ===linear algebra/analytic geometry=== | |||
| The determinant function is defined in terms of vector spaces. It is the only function f: F^n x F^n .... x F^n -> F that is multilinear & alternating such that f( standard basis ) = 1.<br> | |||
| ]/], ], ], <s>] (])</s>, defining a ], ], ], ], ], <s>]</s>, <s>] have same det (])</s>, ], ], ], ] | |||
| Obviously, the above needs a major amount of fleshing out.... | |||
| ===types of matrices=== | |||
| <s>], ], ]</s>, indefinite special orthogonal group, ], | |||
| ], ] | |||
| ===number theory/algebra=== | |||
| ]/]?, ], ]/], ], ], ], ], ] | |||
| ===geometry, analysis=== | |||
| ]?, ], ], <s>]</s>, ], ], ], ], ], absolute value of cx numbers and quaternions (see ]), ] (]), ] | |||
| ===open questions=== | |||
| ], ] | |||
| ===algorithms=== | |||
| ], ], ], <s>], ]</s> | |||
| a few papers: , , , | |||
| ===examples=== | |||
| reflection matrix, <s>],</s> ], ], ] (]), <s>block matrix</s>, ], ], ], det of ] | |||
| ===generalizations=== | |||
| ], ], ], | |||
| <s>], ], ]</s>, <s>det's of infinite matrices / regularized det / ] (see also ]), ]</s>, <s>]</s> | |||
| ===other=== | |||
| ], ], | |||
| ===books=== | |||
| == Include other notions of determinant (of endomorphism and family of vectors) == | |||
| Hello @] and others. | |||
| I would like to at least partially restore my last edit https://en.wikipedia.org/search/?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. | |||
| ] (]) 15:17, 2 October 2024 (UTC) | |||
| :I agree that the structure of the article requires improvements, but your edit goes against the ]: {{tqq|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 {{alink|Definition}}. | |||
| :Also | |||
| :* The heading {{Alink|Abstract algebraic aspects}} must be removed and its subsections must be moved into subsections of {{alink|Generalizations and related notions}}. The right place of this section may be discussed. | |||
| :] (]) 13:48, 3 October 2024 (UTC) | |||
| == Recent removal/readdition == | |||
| {{u|Dominic3203}}, you're continuing to re-add unsourced material about the 2x2 determinant formula, and you're not giving any real substantive reason why it should be included. This article is already too long, and this extra example of a pretty minor formula (already a bit iffy, but whatever) isn't needed. Even if it ''were'' sourced, it's a silly example, as the result is of no apparent importance, being more difficult than just computing the determinant directly. ] (]) 14:20, 8 December 2024 (UTC) | |||
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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
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://en.wikipedia.org/search/?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)
Recent removal/readdition
Dominic3203, you're continuing to re-add unsourced material about the 2x2 determinant formula, and you're not giving any real substantive reason why it should be included. This article is already too long, and this extra example of a pretty minor formula (already a bit iffy, but whatever) isn't needed. Even if it were sourced, it's a silly example, as the result is of no apparent importance, being more difficult than just computing the determinant directly. 35.139.154.158 (talk) 14:20, 8 December 2024 (UTC)
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