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,

Random comment

See math not English adj is baiscly the inevserse of matrix when transposed adj|= inv-1 now o rwos coolums r}^T
Det|adj|=o ~2025-34366-63 (talk) 16:59, 17 November 2025 (UTC)

Sorry, the import of this comment escapes me. Can you be more specific? -- Elphion (talk) 18:04, 17 November 2025 (UTC)

Geometric proof of formula

Proof without words by dissection and rearrangement

@D.Lazard: reverted my addition of this visual proof with this message "Not useful: a professional mathematician needs several minutes for understanding what is meant". I think a visual explanation complements the algebraic interpretation. How can the diagram be improved to be clearer? Thanks, cmɢʟee τaʟκ (please add {{ping|cmglee}} to your reply) 18:31, 28 July 2026 (UTC)

To editor Cmglee: Sorry, a figure that requires several minutes for being understood by a professional mathematician cannot be called a "visual explanation". IMO, there is no hope to improve the figure for getting something acceptable. In any case, without any similar figure in most common textbooks dealing with determinants, there is no hope to not fall under the policy WP:No original research. D.Lazard (talk) 20:58, 28 July 2026 (UTC)

It might work as a 2-panel illustration, showing first the pieces in place in the parallelogram, and second their rearrangement, united by the visual equation at the bottom. Then leave the curved arrows out -- the colors take care of showing where the pieces go. (The "no original research" argument is nonsense; this is illustration, not research. Extending NOR to situations like this would require eliminating most of the math diagrams on WP.) -- Elphion (talk) 21:13, 28 July 2026 (UTC)

Thanks for your feedback, @D.Lazard: and @Elphion:.

It's based on http://i.sstatic.net/gCaz3.png by Solomon W. Golomb (appeared in Mathematics Magazine, March 1985), simplified to avoid overlapping areas.

I'll update it to two panels and remove the curved arrows. I'll ping you when it's ready.

Cheers, cmɢʟee τaʟκ (please add {{ping|cmglee}} to your reply) 22:17, 28 July 2026 (UTC)

I find the original easier to follow, though still pretty confusing, among the less obvious of "proofs without words" I've seen. I think all of the arrows and extra labels in the new versions are more distracting than helpful. –jacobolus (t) 00:54, 29 July 2026 (UTC)