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Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations represented by them.

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Overview

Two by two matrices

Geometric meaning

Definition

Examples

Properties

Properties of the determinant in relation to other notions

History

Applications

Abstract algebraic aspects

Berezin integral

Generalizations and related notions

Calculation

Historical references

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Determinant

Nodes303
Edges302
Triples421
Avg. degree1.99
Density0.006601
Components1

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Determinant

Top relations

related to References · 130
Determinant → Abstract, Abstract Algebra, ACM SIGNUM Newsletter, Addison Wesley, Algebra, American Mathematical Monthly, An, Anton, Applications Version, Applied Linear Algebra, Applied Mathematics, Archived, Arel, August, Baley Price, Bareiss, Bau III, Birkhäuser, Boor, Brooks/Cole
related to Historical references · 61
Determinant → Académie Royale, An, Analysis, Annals, Arthur, Bourbaki, Cambridge Mathematical Journal, Cambridge University PressE, Carl Gustav Jakob, Cl, Companion Encyclopedia, De Determinantibus, Determinants, Dover, Elements, Florian, Forsyth Scott, Frères Cramer, Gabriel, Genève
related to External links · 23
Determinant → Archived, Determinant Calculator Calculator, Earliest Uses PagesDeterminants, Edmund, EMS PressWeisstein, Encyclopedia, Eric, John, Laplace, Linear Algebra, MacTutor History, Mathematics, Mathematics Archive, MathWorld, Matrices, O'Connor, Robertson, St AndrewsDeterminant Interactive Program, Suprunenko, TutorialLinear
has method · 22
Determinant → Algorithms, Bareiss Algorithm, By, Cholesky, Coppersmith, Especially, Examples, For, Gauss, Gaussian, If, In, Leibniz, LU, One, QR, Some, Such, The, These
is a · 16
Determinant → coefficient of the standard volume element e 1, concave function, functional determinant, functional determinant.Operators in von Neumann algebrasFor operators in a finite factor, homogeneous function, homogeneous function.Sum identity for 2, invertible element in R, morphism of algebraic groups, multiplicative map, n-linear function, n-linear function.Multiplicativity and matrix groupsThe determinant is a multiplicative map, natural transformation between the two functors GL n, product of the entries of its diagonal.So, resultant, scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations represent…, scale factor by which the transformation alters every volume in the space
related to history · 15
Determinant → Both Cramer, Bourbaki, Bézout, Cardano, Cramer, Cramer's, Determinants, Grassmann, Historically, In Europe, Japan, Lang, Leibniz, Seki Takakazu, The
related to Immediate consequences · 13
Determinant → Adding, Alternatively, As, For, Given, If, In, Indeed, Interchanging, Leibniz, The, These, This
related to Related notions for non-commutative rings · 9
Determinant → Berezinian, Capelli, Dieudonné, Examples, For, Leibniz, Manin, Nevertheless, Over
related to Calculation · 8
Determinant → Computational, Determinants, Leibniz, The Laplace, Therefore, They, Thus, While
related to Linear independence · 8
Determinant → Another, Determinants, For, If, It, See, The, Wronskian

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Important terminology

displaystyle matrix matrices determinants linear entries det formula times one columns leibniz also two column terms algebra product used using

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Determinantis ascalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations represent…0.90text
Determinantis ascale factor by which the transformation alters every volume in the space0.90text
Determinantis acoefficient of the standard volume element e 10.90text
Determinantis ahomogeneous function0.90text
Determinantis an-linear function.Multiplicativity and matrix groupsThe determinant is a multiplicative map0.90text
Determinantis aconcave function0.90text
Determinantis ahomogeneous function.Sum identity for 20.90text
Determinantis an-linear function0.90text
Determinantis amultiplicative map0.90text
Determinantis aresultant0.90text
Determinantis ainvertible element in R0.90text
Determinantis anatural transformation between the two functors GL n0.90text
Determinantis amorphism of algebraic groups0.90text
Determinantis afunctional determinant.Operators in von Neumann algebrasFor operators in a finite factor0.90text
Determinantis afunctional determinant0.90text
Determinantis aproduct of the entries of its diagonal.So0.90text

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