Talk:Determinant

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Right handed coordinante[edit]

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[edit]

linear algebra/analytic geometry[edit]

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[edit]

special linear group, special orthogonal group, special unitary group, indefinite special orthogonal group, modular group, unimodular matrix, matrices with multidimensional indices

number theory/algebra[edit]

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[edit]

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[edit]

Jacobian conjecture, Hadamard's maximal determinant problem

algorithms[edit]

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[edit]

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[edit]

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[edit]

Determinantal point process, Kirchhoff's theorem,

books[edit]

[1]

physical meaning of determinant[edit]

Let suppose if we have a vector which make 2x2 matrix and if we take its determinant then this value will represent its magnitude or something else? What are the meaning of determinant? Please explain it using vectors Muzammalsafdar (talk) 15:55, 19 April 2016 (UTC)

Product of scaling factors (eigenvalues) of its eigenvectors. May wish to go to eigenvalue and eigenvector article. This is the wrong place. Here, the physical connection to areas and volumes is expounded and illustrated in sections 1.1, 1.2, 8.3. Cuzkatzimhut (talk) 19:31, 19 April 2016 (UTC)

Simple proof of inequality [edit]

Let λi are positive eigenvalues of A. Inequality 1 - 1/λi ≤ ln(λi) ≤ λi - 1 is known from standard courses in math. By taking the sum over i we obtain Σi(1 - 1/λi) ≤ ln(Πiλi) ≤ Σii - 1). In terms of trace and determinant functions, tr(I - Λ-1) ≤ ln(det(Λ)) ≤ tr(Λ - I), where Λ = diag(λ12,...,λn). Substituting Λ = UAU-1 and eliminating U, we obtain the inequality tr(I - A-1) ≤ ln(det(A)) ≤ tr(A - I). Trompedo (talk) 12:00, 17 July 2016 (UTC)

You may wish to insist on strictly positive-definite matrix A. Cuzkatzimhut (talk) 14:45, 17 July 2016 (UTC)