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Quaternion

In mathematics, the quaternions form a number system similar to the complex numbers, with the usual arithmetical operations of addition, subtraction, multiplication, and division, but with four real-number components instead of two. Unlike with the complex numbers, quaternion multiplication is not commutative, meaning that the result of multiplying two…

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Overview

History

Definition

Conjugation, the norm, and reciprocal

Algebraic properties

Quaternions and three-dimensional geometry

Matrix representations

Lagrange's four-square theorem

Quaternions as pairs of complex numbers

Square roots

Functions of a quaternion variable

Three-dimensional and four-dimensional rotation groups

Quaternions as the even part of Cl3,0(ℝ)

Brauer group

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Map overview Semantic statistics

Quaternion

Nodes243
Edges242
Triples138
Avg. degree1.99
Density0.00823
Components1

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Quaternion

Top relations

see also · 15
Quaternion → Algebra, Concept, Conversion, Eight-dimensional, Euclidean, Euler, Four-dimensional, Function, Functions, Mathematical, Matrices, Mutation, Quaternions, Special, Transparent
related to Algebraic properties · 11
Quaternion → Banach, Because, Even, Finite-dimensional, Hurwitz's, Multiplication, The, The Frobenius, Therefore, This, Those
related to history · 11
Quaternion → Euler's, Ferdinand Georg Frobenius, Gauss, Hamilton, He, However, Important, In, Olinde Rodrigues, Points, Quaternions
related to Quaternions in physics · 11
Quaternion → As, Cornelius Lanczos, Hamilton, Lectures, Lorentz, Ludwik Silberstein, Pauli, Plate, Quaternions, The, This
related to External links · 10
Quaternion → Archive, Formal, Formal Proofs, Isabelle/HOL, Lawrence, Media, Quaternions, Visualisation, Wikimedia CommonsPaulson, Wiktionary-logo-en-v2
related to Brauer group · 9
Quaternion → Artin, Brauer, By, CSA, CSAs, Explicitly, The, Wedderburn, Wedderburn's
related to Quaternions as pairs of complex numbers · 7
Quaternion → Cayley, Choose, Dickson, From, Let, Quaternions, This
related to Conjugation, the norm, and reciprocal · 6
Quaternion → Clifford, Conjugation, It, That, The, To
related to Functions of a quaternion variable · 6
Quaternion → Examples, For, Julia, Like, Mandelbrot, Maxwell
related to Lagrange's four-square theorem · 6
Quaternion → As, Euclidean, Hurwitz, Lagrange's, Quaternions, The

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Important terminology

quaternions displaystyle real vector numbers mathbb algebra complex mathbf two basis matrices also space multiplication scalar one unit part group

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Quaternionis aexpression of the form a0.90text
Quaternionis azero quaternion0.90text
Quaternionis aquaternion of norm one0.90text
Quaternionis adeterminant of the corresponding matrix0.90text
the vector dotinstance ofOperations0.80text
cross products can be defined in terms of quaternionsinstance ofOperations0.80text
and this makes it possible to apply quaternion techniques wherever spatial vectors ariseinstance ofOperations0.80text
Euler angles.Fasterinstance ofa problem with systems0.80text
more compact than matrices.Nonsingular representationinstance ofa problem with systems0.80text
Quaternionrelated to Algebraic propertiesThe0.60section
Quaternionrelated to Algebraic propertiesMultiplication0.60section
Quaternionrelated to Algebraic propertiesTherefore0.60section

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