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Concept in linear algebra
A quaternionic matrix is a matrix whose elements are quaternions. The quaternions form a noncommutative ring, and therefore addition and multiplication
Quaternionic_matrix
Representation of a group or algebra in terms of an algebra with quaternionic structure
a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and
Quaternionic_representation
Four-dimensional number system
Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective
Quaternion
Concept in mathematics
quasideterminant Moore, E. H. (1922), "On the determinant of an hermitian matrix with quaternionic elements. Definition and elementary properties with applications
Moore determinant of a Hermitian matrix
Moore_determinant_of_a_Hermitian_matrix
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_matrix
transformations that arise by left-multiplication by some quaternionic n × n {\displaystyle n\times n} matrix, while the group S p ( 1 ) = S 3 {\displaystyle Sp(1)=S^{3}}
Quaternion-Kähler_manifold
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Matrix-valued random variable
ensembles. Invariant matrix ensembles are random Hermitian matrices with density on the space of real symmetric/Hermitian/quaternionic Hermitian matrices
Random_matrix
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Hypercomplex number system
}^{*}\end{bmatrix}}} Using a slightly modified (non-associative) quaternionic matrix multiplication: [ α 0 α 1 α 2 α 3 ] ∘ [ β 0 β 1 β 2 β 3 ] = [ α 0
Octonion
Function theory with quaternion variable
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of
Quaternionic_analysis
Element of a unital algebra over the field of real numbers
{\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields a quaternionic matrix and its even subalgebra H ⊗ 2 ⊗ R C {\displaystyle \mathbb {H} ^{\otimes
Hypercomplex_number
Concept in mathematics
determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism
Moore_matrix
Random matrix with gaussian entries
only 3 real division algebras: the real, the complex, and the quaternionic. A random matrix representing a Hamiltonian H {\displaystyle H} can be classified
Gaussian_ensemble
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
\mathbb {CP} ^{n}} with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf
Hopf_fibration
Mathematical operation
transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a
Cayley_transform
Branch of mathematical analysis
quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis). A second instance involves functions of a motor variable where
Hypercomplex_analysis
Correspondence between quaternions and 3D rotations
{\displaystyle {\vec {u}}} that specifies a rotation as to axial vectors. In quaternionic formalism the choice of an orientation of the space corresponds to order
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Non-tensorial representation of the spin group
conditions. When S {\displaystyle S} is of quaternionic type, the representation carries an invariant quaternionic structure but no invariant real structure
Spinor
Type of group in mathematics
traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups
Classical_group
Canadian-American mathematician
1994 Thompson, Robert C. (1997). "The upper numerical range of a quaternionic matrix is not a complex numerical range". Linear Algebra and Its Applications
Robert_Charles_Thompson
Mathematical group
\operatorname {Sp} (n)} is given by the quaternionic skew-Hermitian matrices, the set of n × n {\displaystyle n\times n} quaternionic matrices that satisfy A + A
Symplectic_group
Mathematical object
quaternion; that is, a quaternion that satisfies τ2 = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie
3-sphere
matrices, and the circular symplectic ensemble (CSE) on self dual unitary quaternionic matrices. The distribution of the unitary circular ensemble CUE(n) is
Circular_ensemble
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
orthonormal k-frames in C n {\displaystyle \mathbb {C} ^{n}} and the quaternionic Stiefel manifold V k ( H n ) {\displaystyle V_{k}(\mathbb {H} ^{n})}
Stiefel_manifold
Type of Riemannian manifold
respect to the Riemannian metric g {\displaystyle g} and satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − 1 {\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}
Hyperkähler_manifold
On eigenvalues of random matrices
all large n {\displaystyle n} . The theorem still holds for quaternionic non-Hermitian matrix ensembles, with e − e − x {\displaystyle e^{-e^{-x}}} replaced
Circular_law
American physicist
Routines, World Scientific Publishing Co., 2012, ISBN 978-981-4425-03-2 Quaternionic Quantum Mechanics and Quantum Fields, International Series of Monographs
Stephen_L._Adler
Classification in abstract algebra
whether the relevant central simple algebra is split or quaternionic. In even dimension this yields matrix algebras over R {\displaystyle \mathbf {R} } or H
Classification of Clifford algebras
Classification_of_Clifford_algebras
Regular object in four dimensional geometry
belongs to 4 hexagons, and each hexagon contains 3 axes. This configuration matrix represents the 24-cell. The rows and columns correspond to vertices, edges
24-cell
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
counter-example are the special orthogonal groups in even dimension. These have the matrix − I {\displaystyle -I} in the center, and this element is path-connected
Simple_Lie_group
Four-dimensional associative algebra over the reals
2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature, k is replaced
Split-quaternion
Method of constructing instanton solutions
Let x be the 4-dimensional Euclidean spacetime coordinates written in quaternionic notation x i j = ( z 2 z 1 − z 1 ¯ z 2 ¯ ) . {\displaystyle
ADHM_construction
Particular projective representations of the orthogonal or special orthogonal groups
that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure. There is an invariant complex antilinear
Spin_representation
Finite simple group type not classified as Lie, cyclic or alternating
a type 2-3-3 triangle J2 is the group of automorphisms preserving a quaternionic structure (modulo its center). Consists of subgroups which are closely
Sporadic_group
geometry used to describe the physical phenomena of quantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Fundamental construction of differential calculus
derivative corresponds to the integral, whence the term differintegral. In quaternionic analysis, derivatives can be defined in a similar way to real and complex
Generalizations of the derivative
Generalizations_of_the_derivative
Spin representations of the SO(3) group
3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that X is a matrix representing a vector x in
Spinors_in_three_dimensions
Geometric concept of a 2D space with "points at infinity" adjoined
pappian planes) serve as fundamental examples in algebraic geometry. The quaternionic projective plane HP2 is also of independent interest. By Wedderburn's
Projective_plane
Mnemonic for 3D vectors orientations and rotations
Following a substantial debate, the mainstream shifted from Hamilton's quaternionic system to Gibbs's three-vectors system. This transition led to the prevalent
Right-hand_rule
Quaternions with complex number coefficients
Complex Quaternions and Maxwell's Equations. Furey 2012. L. Silberstein, Quaternionic Form of Relativity, Philos. Mag. S., 6, Vol. 23, No. 137, pp. 790-809
Biquaternion
structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler. Hopf, Heinz (1948), "Zur Topologie
Hopf_manifold
Group of unitary matrices
{\displaystyle n\times n} unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group
Unitary_group
Supergravity in eleven dimensions
squashed 7-sphere, which can be acquired by embedding the 7-sphere in a quaternionic projective space, with this giving a gauge group of SO ( 5 ) × SU ( 2
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
Mathematical concept
diffeomorphic to the sphere, or isometric to the complex projective space, the quaternionic projective space, or else the Cayley plane F4/Spin(9); see (Brendle &
Complex_projective_space
Fringe theory of physics
single Lie group geometry—specifically, excitations of the noncompact quaternionic real form of the largest simple exceptional Lie group, E8. A Lie group
An Exceptionally Simple Theory of Everything
An_Exceptionally_Simple_Theory_of_Everything
Riemannian metric on the space of mixed states of a quantum system
after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the
Bures_metric
Smooth manifold with an inner product on each tangent space
metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues of the real projective
Riemannian_manifold
Four finite groups derived from the Leech lattice
Hall–Janko group J2 (order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars. The seven simple groups
Conway_group
288-cell is the only non-regular 4-polytope which is the convex hull of a quaternionic group, disregarding the infinitely many dicyclic (same as binary dihedral)
Truncated_24-cells
v+W\mapsto gv+W} . quaternionic A quaternionic representation of a group G is a complex representation equipped with a G-invariant quaternionic structure. quiver
Glossary of representation theory
Glossary_of_representation_theory
Algebraic structure designed for geometry
analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis
Geometric_algebra
Equations describing classical electromagnetism
the geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used. Maxwell's equations
Maxwell's_equations
Mathematics term
≥ 2. For n ≥ 2, the noncompact Lie group Sp(n, 1) of isometries of a quaternionic hermitian form of signature (n,1) is a simple Lie group of real rank
Kazhdan's_property_(T)
Every polynomial has a real or complex root
Eilenberg–Niven theorem, a generalization of the theorem to polynomials with quaternionic coefficients and variables Hilbert's Nullstellensatz, a generalization
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Type of group representation for locally compact groups
functors. Blattner's conjecture Holomorphic discrete series representation Quaternionic discrete series representation Atiyah, Michael; Schmid, Wilfried (1977)
Discrete series representation
Discrete_series_representation
Manifold
first Chern class vanishes. Complex dimension Complex analytic variety Quaternionic manifold Real-complex manifold One must use the open unit ball in the
Complex_manifold
Classification system for symmetry groups in geometry
Commutator subgroup, p. 124–126 Johnson, Norman W.; Weiss, Asia Ivić (1999). "Quaternionic modular groups". Linear Algebra and Its Applications. 295 (1–3): 159–189
Coxeter_notation
Study of complex manifolds and several complex variables
complex structures I , J , K {\displaystyle I,J,K} which satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − Id {\displaystyle
Complex_geometry
Theory of supergravity in four dimensions
{\mathcal {N}}=2} supergravity the relevant scalar manifold must be a quaternionic Kähler manifold. But since these manifolds are not themselves Kähler
4D_N_=_1_supergravity
Special mathematical functions defined on the surface of a sphere
certain spin representations of SO(3), with respect to the action by quaternionic multiplication. Spherical harmonics can be separated into two sets of
Spherical_harmonics
Complex vector of electromagnetic fields
transition is made: With the advent of spinor calculus that superseded the quaternionic calculus, the transformation properties of the Riemann-Silberstein vector
Riemann–Silberstein_vector
Special type of principal bundle
four-dimensional sphere S 4 {\displaystyle S^{4}} , which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles
Principal_SU(2)-bundle
U(N) to U(N – 1) states that Example. The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations
Restricted_representation
In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When n = 4, the Dirac operator is sometimes referred to as
Clifford_analysis
Four-dimensional analog of the dodecahedron
S2CID 119288632. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8
120-cell
Concept in mathematics
indicates whether a given irreducible character is real, complex, or quaternionic. They are examples of Schur functors. They are defined as follows. Let
Tensor product of representations
Tensor_product_of_representations
Geometric model of the physical space
5. ISBN 978-0-19-960139-4. Morais, João Pedro; et al. (2014). Real Quaternionic Calculus Handbook. Springer Science & Business Media. pp. 1–13. ISBN 978-3-0348-0622-0
Three-dimensional_space
Development of linear transformations forming the Lorentz group
2}+C^{\prime 2}+D^{\prime 2}\end{aligned}}\end{matrix}}} Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic
History of Lorentz transformations
History_of_Lorentz_transformations
Generalization of the concept of directional derivative
Generalization of a derivative of a function between two Banach spaces Quaternionic analysis – Function theory with quaternion variable Semi-differentiability –
Gateaux_derivative
Theorem in quantum mechanics
measurements are defined must be a real or complex Hilbert space, or a quaternionic module. (Gleason's argument is inapplicable if, for example, one tries
Gleason's_theorem
Structure group sub-bundle on a tangent frame bundle
a system of basis-dependent 1-forms ω via ∇X Vi = ωij(X)Vj where, as a matrix of 1-forms, ω ∈ Ω1(M)⊗gl(n). An adapted connection is one for which ω takes
G-structure_on_a_manifold
Mathematical result in differential geometry
that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so
Atiyah–Singer_index_theorem
Representations of finite groups, particularly on vector spaces
complex conjugate representations of G . {\displaystyle G.} Definition. A quaternionic representation is a (complex) representation V , {\displaystyle V,} which
Representation theory of finite groups
Representation_theory_of_finite_groups
Metric on a complex projective space endowed with Hermitian form
coordinates, one then defines polar coordinate one-forms on the 4-sphere (the quaternionic projective line) as r d r = + x d x + y d y + z d z + t d t r 2 σ 1 =
Fubini–Study_metric
Lie groups and their associated Lie algebras
JA + ATJ = 0 where J is the standard skew-symmetric matrix Yes Yes n(2n+1) sp(n) square quaternionic matrices A satisfying A = −A∗, with Lie bracket the
Table_of_Lie_groups
Generalized sphere of dimension n (mathematics)
-sphere, Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic projective line, H P 1 {\displaystyle \mathbf {HP} ^{1}} . SO
N-sphere
Generalization of a polytope in real space
triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection. Quaternionic polytope Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation
Complex_polytope
Low-rank isomorphisms in mathematics
)\times \mathrm {SL} (2,\mathbf {R} )\to \mathrm {SO} (2,2).} On the quaternionic real form one recovers the compact case S U ( 2 ) × S U ( 2 ) → S O (
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
Completion of the usual space with "points at infinity"
naturally to the case where K is a division ring; see, for example, Quaternionic projective space. The notation PG(n, K) is sometimes used for Pn(K).
Projective_space
ISBN 978-1-56881-220-5. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8
Dual_snub_24-cell
Form of differential geometry
the quaternionic projective plane is not its systolically optimal metric, in contrast with the 2-systole in the complex case. While the quaternionic projective
Systolic_geometry
Mathematical concept
Sabadini; M Shapiro; F Sommen (eds.). Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed.). Birkhäuser. p. 168. ISBN 978-3-7643-9892-7
Seven-dimensional cross product
Seven-dimensional_cross_product
Representation theory
the Weyl group of A. The group G = SL(2,C) acts transitively on the quaternionic upper half space H 3 = { x + y i + t j ∣ t > 0 } {\displaystyle {\mathfrak
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial
List_of_women_in_mathematics
Four-dimensional analog of the icosahedron
Cartesian coordinate — the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group
600-cell
QUATERNIONIC MATRIX
QUATERNIONIC MATRIX
Biblical
a guard of four soldiers,...and delivered him to four quaternions of soldiers to guard him...
QUATERNIONIC MATRIX
QUATERNIONIC MATRIX
Surname or Lastname
English
English : habitational name from either of two places in North Yorkshire called Helmsley. The names are of different etymologies: the one near Rievaulx Abbey is from the Old English personal name Helm + Old English lēah ‘wood’, ‘clearing’, whereas Upper Helmsley, near York, is from the Old English personal name Hemele + Old English ēg ‘island’, and had the form Hemelsey till at least the 14th century.
Surname or Lastname
English
English : habitational name from either of two places, in Staffordshire and Sussex, named Betley, from an Old English female personal name Bette + lēah ‘woodland clearing’.
Boy/Male
Hebrew
Heroic.
Boy/Male
Arabic, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Sanskrit, Tamil, Telugu
Natural; Original; Easy
Boy/Male
American, Basque, French, Hindu, Indian, Latin, Spanish
Judicious; Counsel; Advice; Wise; Famous
Boy/Male
Norse
Wolfs friend.
Surname or Lastname
English and Irish
English and Irish : from a pet form of the medieval male personal name Stace, a reduced vernacular form of Eustace.
Surname or Lastname
English
English : nickname from the animal, Middle English, Old English fox. It may have denoted a cunning individual or been given to someone with red hair or for some other anecdotal reason. This relatively common and readily understood surname seems to have absorbed some early examples of less transparent surnames derived from the Germanic personal names mentioned at Faulks and Foulks.Irish : part translation of Gaelic Mac an tSionnaigh ‘son of the fox’ (see Tinney).Jewish (American) : translation of the Ashkenazic Jewish surname Fuchs.Americanized spelling of Focks, a North German patronymic from the personal name Fock (see Volk).Americanized spelling of Fochs, a North German variant of Fuchs, or in some cases no doubt a translation of Fuchs itself.
Boy/Male
Hindu, Indian
Rice Put During Pooja in Hindu Rituals
Girl/Female
Indian
Melody, Song
QUATERNIONIC MATRIX
QUATERNIONIC MATRIX
QUATERNIONIC MATRIX
QUATERNIONIC MATRIX
QUATERNIONIC MATRIX
v. t.
To divide into quaternions, files, or companies.
n.
See Matrix.
n.
A mold; a matrix.
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
v. t.
The white fibrous matter forming the matrix from which fungi.
n.
The number four; a collection of four things; a quaternion.
n.
The number four.
n.
Hence, that which gives form or origin to anything
n.
A word of four syllables; a quadrisyllable.
n.
In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.
pl.
of Matrix
n.
The turning factor of a quaternion.
n.
A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.
n.
The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
n.
The cavity in which anything is formed, and which gives it shape; a die; a mold, as for the face of a type.
n.
The amorphous or homogenous matrix or ground mass, as distinguished from well-defined crystals; as, the magma of porphyry.
n.
The womb.
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.