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algebra, the quaternionic eigenvalue problem is the problem of determining the eigenvalues and eigenvectors of a matrix with quaternionic entries. Unlike
Quaternionic eigenvalue problem
Quaternionic_eigenvalue_problem
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
Matrix-valued random variable
as a random matrix. For larger atoms, the distribution of the energy eigenvalues of the Hamiltonian could be computed in order to approximate scattering
Random_matrix
Random matrix with gaussian entries
{\displaystyle M^{*}} is its transpose. If M {\displaystyle M} is complex or quaternionic, then M ∗ {\displaystyle M^{*}} is its conjugate transpose. λ 1 , …
Gaussian_ensemble
Matrix equal to its conjugate-transpose
Ritz and Lord Rayleigh. Parlett, Beresford N. (1998). The symmetric eigenvalue problem. Classics in applied mathematics. Philadelphia, Pa: Society for Industrial
Hermitian_matrix
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
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)
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
Matrix representing a Euclidean rotation
eigenvector of R corresponding to the eigenvalue λ = 1. Every rotation matrix must have this eigenvalue, the other two eigenvalues being complex conjugates of each
Rotation_matrix
. Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices
List_of_named_matrices
Type of Riemannian manifold with constant Jacobi operator spectrum
{\displaystyle \mathbb {CH} ^{n}} , quaternionic projective spaces H P n {\displaystyle \mathbb {HP} ^{n}} , quaternionic hyperbolic spaces H H n {\displaystyle
Osserman_manifold
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
Special mathematical functions defined on the surface of a sphere
φ), Y : S 2 → C {\displaystyle Y:S^{2}\to \mathbb {C} } , of the eigenvalue problem r 2 ∇ 2 Y = − ℓ ( ℓ + 1 ) Y {\displaystyle r^{2}\nabla ^{2}Y=-\ell
Spherical_harmonics
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
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
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
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
Russian-French mathematician
Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978, Gromov introduced the
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
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
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
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
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
Spin representations of the SO(3) group
constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that
Spinors_in_three_dimensions
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
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
researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial
List_of_women_in_mathematics
Type of Dirichlet series associated to number field extensions
algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root number is the subject of significant research
Artin_L-function
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
Discrete subgroup in a locally compact topological group
1)} (groups of matrices with quaternion coefficients which preserve a "quaternionic quadratic form" of signature ( n , 1 ) {\displaystyle (n,1)} ) for n
Lattice_(discrete_subgroup)
QUATERNIONIC EIGENVALUE-PROBLEM
QUATERNIONIC EIGENVALUE-PROBLEM
Girl/Female
Indian, Telugu
Destroyer of Problems
Boy/Male
Arabic, Indian, Muslim
Problem Solver
Girl/Female
Bengali, Indian
Eternity; Problem Solver
Boy/Male
Muslim
Problem solver
Boy/Male
Indian, Tamil
People with this Name are Preferably Intelligent and Very Generous; Highly Knowledgeable in Problem Solving Skills
Girl/Female
Muslim/Islamic
Away from all Problems
Biblical
a guard of four soldiers,...and delivered him to four quaternions of soldiers to guard him...
Boy/Male
Hindu, Indian
Problem
QUATERNIONIC EIGENVALUE-PROBLEM
QUATERNIONIC EIGENVALUE-PROBLEM
Boy/Male
Indian, Punjabi, Sikh
Most Excellent
Girl/Female
Hindu, Indian
Downward
Boy/Male
Hindu
King of the world
Female
Romanian
Romanian name CRINA means "lily."
Boy/Male
Arabic, Muslim, Parsi
Adorning / Crown of the Religion (Islam)
Girl/Female
German
Noble she wolf.
Boy/Male
Biblical
Tent, tabernacle, brightness.
Boy/Male
Arabic, Muslim, Sindhi
Agency; Advocacy
Surname or Lastname
English
English : from an old spelling of Oxford.
Boy/Male
Indian, Sanskrit
Lord of Nobles; Husband of Noble Family
QUATERNIONIC EIGENVALUE-PROBLEM
QUATERNIONIC EIGENVALUE-PROBLEM
QUATERNIONIC EIGENVALUE-PROBLEM
QUATERNIONIC EIGENVALUE-PROBLEM
QUATERNIONIC EIGENVALUE-PROBLEM
v. t.
To propose problems.
n.
In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.
n.
The number four; a collection of four things; a quaternion.
v. t.
To divide into quaternions, files, or companies.
a.
Having the nature of a problem; not shown in fact; questionable; uncertain; unsettled; doubtful.
v. t.
To explain; to resolve; to unfold; to clear up (what is obscure or difficult to be understood); to work out to a result or conclusion; as, to solve a doubt; to solve difficulties; to solve a problem.
n.
A problem to be solved, or an example to be wrought out.
n.
The turning factor of a quaternion.
n.
To cause to stick; to bring to a stand; to pose; to puzzle; as, to stick one with a hard problem.
n.
One who proposes problems.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
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.
a.
Alt. of Problematical
a.
Questionable; equivocal; indefinite; problematical.
n.
The number four.
n.
To begin to deal with; as, to tackle the problem.
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.
A problem of more than usual difficulty added to another on an examination paper.
n.
A word of four syllables; a quadrisyllable.
v. t.
To have just and adequate ideas of; to apprehended the meaning or intention of; to have knowledge of; to comprehend; to know; as, to understand a problem in Euclid; to understand a proposition or a declaration; the court understands the advocate or his argument; to understand the sacred oracles; to understand a nod or a wink.