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Linear operator used in quantum mechanics
In physics, particularly in quantum perturbation theory, the matrix element refers to the linear operator of a modified Hamiltonian using Dirac notation
Matrix_element_(physics)
Topics referred to by the same term
Matrix element may refer to: The (scalar) entries of a matrix. Matrix element (physics), the value of a linear operator (especially a modified Hamiltonian)
Matrix_element
Matrix operation generalizing exponentiation of scalar numbers
evaluated at t = 1. If D is an n × n diagonal matrix, exp(D) is an n × n diagonal matrix with each diagonal element equal to the ordinary exponential function
Matrix_exponential
Distinguished element of a Lie algebra's center
adjoint action of any element as a 3×3 matrix. In particular: 𝖠𝖽e=[0−20001000],𝖠𝖽f=[000−100020],𝖠𝖽h=[20000000−2] The Casimir element then acts on this
Casimir_element
Matrix equal to its conjugate-transpose
matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose. That is, if the element in
Hermitian_matrix
Array of numbers
[19−13205−6] denotes a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3
Matrix_(mathematics)
Square matrix used to represent a graph or network
set U = {u1, ..., un}, the adjacency matrix is a square n × n matrix A such that its element Aij is 1 when there is an edge from vertex ui to vertex uj,
Adjacency_matrix
Matrices important in quantum mechanics and the study of spin
traceless matrix-valued dual vector, that is, an element of Mat2×2(ℂ)⊗(ℝ3)∗ that maps 𝒂↦𝒂⋅𝝈 Each component of 𝒂 can be recovered from the matrix (see
Pauli_matrices
Form of a matrix
diagonal elements of a skew-symmetric matrix are zeros because each element must be its own negative. The matrix A=[02−45−20−44540] is skew-symmetric because
Skew-symmetric_matrix
Matrix representing the effect of scattering on a physical system
In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a scattering
S-matrix
Irreducible representation of the rotation group SO
β,0) is an element of the orthogonal Wigner's (small) d-matrix (sometimes referred to as the reduced Wigner D-matrix). That is, in this basis
Wigner_D-matrix
Numerical method for solving physical or engineering problems
dynamic implicit and explicit finite element simulations of the native knee joint" (PDF). Medical Engineering & Physics. 38 (10): 1123–1130. doi:10.1016/j
Finite_element_method
Matrix operation which flips a matrix over its diagonal
column element of AT is the jth row, ith column element of A: [𝐀T]ij=[𝐀]ji. If A is an m × n matrix, then AT is an n × m matrix. A square matrix whose
Transpose
Ray tracing technique
paraxial rays. Each optical element (surface, interface, mirror, or beam travel) is described by a 2 × 2 ray transfer matrix which operates on a vector
Ray_transfer_matrix_analysis
System for describing optical polarization
particular optical element is represented by a Mueller matrix—a 4×4 matrix that is an overlapping generalization of the Jones matrix. Disregarding coherent
Mueller_calculus
Topics referred to by the same term
Unitary element Unitary group Unitary matrix Unitary morphism Unitary operator Unitary transformation Unitary representation Unitarity (physics) E-unitary
Unitary
Formulation of quantum mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually
Matrix_mechanics
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
Matrix relating a system's generalized coordinate vector and kinetic energy
deformation of each element is ignored, creates a diagonal mass matrix and negates the need to integrate mass across the deformed element. Moment of inertia
Mass_matrix
Mathematical operation in linear algebra
as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational
Matrix_multiplication
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
Rotation_matrix
Specialized notation for multivariable calculus
tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The
Matrix_calculus
Most widely known generalized inverse of a matrix
a matrix, but sometimes applied to other elements of algebraic structures which share some but not all properties expected for an inverse element. A
Moore–Penrose_inverse
Matrix describing continuous-time Markov chains
between states. In a transition-rate matrix Q {\displaystyle Q} (sometimes written A {\displaystyle A} ), element q i j {\displaystyle q_{ij}} (for i ≠
Transition-rate_matrix
Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
(often in physics) 𝐀†. For real matrices, the conjugate transpose is just the transpose, 𝐀H=𝐀T. The conjugate transpose of an m×n matrix 𝐀 is formally
Conjugate_transpose
Vector formula for a rotation in space, given its axis
the rotation matrix through an angle θ counterclockwise about the axis k, and I the 3 × 3 identity matrix. This matrix R is an element of the rotation
Rodrigues'_rotation_formula
Gamma matrices for arbitrary Clifford algebras
T-symmetry in physics; all directions are reversed. Define the chiral element ω≡Γchir as ω=Γchir=Γ0Γ1⋯Γd−1 where d=p+q. The chiral element commutes with
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Matrix-valued random variable
probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries
Random_matrix
centroid of each element. The 2D element stiffness matrix size is 6 × 6; the components of the upper left quarter of the stiffness matrix are shown below:
Applied_element_method
Vector operation
two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two
Outer_product
Line segment of infinitesimally small length
definition of the square of the line element ds in an n-dimensional Riemannian or pseudo-Riemannian manifold (in physics usually a Lorentzian manifold) is
Line_element
Four-dimensional number system
Mathematics, EMS Press, 2001 [1994] "Matrix and quaternion". Frequently Asked Questions. 1.21. Sweetser, Doug. "Doing physics with quaternions". Hoffman, Gernot
Quaternion
Mathematical operation on vectors in 3D space
cross points to in the right-hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as
Cross_product
Matrix whose only nonzero elements are on its main diagonal
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices
Diagonal_matrix
Matrix used to describe the transitions of a Markov chain
It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov
Stochastic_matrix
engineering software List of finite element software packages List of mathematical software List of physicists List of plasma physics software List of quantum chemistry
List of computational physics software
List_of_computational_physics_software
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
Intrinsic quantum property of particles
first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as SymPy; while many physics textbooks
Spin_(physics)
Family of linear transformations
neighborhoods of the identity element of the Lie group. In the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the
Lorentz_transformation
Matrices similar to diagonal matrices
square n×n matrix A with entries in a field F is called diagonalizable or nondefective if there exists an n×n invertible matrix (i.e. an element of the general
Diagonalizable_matrix
Branch of physics
banded matrix inversion to calculate the weights of basis functions (when modeled by finite element methods); matrix products (when using transfer matrix methods);
Computational electromagnetics
Computational_electromagnetics
Element in a ring whose some power is 0
nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). All nilpotent elements are zero divisors. An n×n matrix A
Nilpotent
Function approximating net physical effect
without including all of the underlying physics, but instead, providing the momentum dependence of suitable matrix elements. It is further measured experimentally
Form factor (quantum field theory)
Form_factor_(quantum_field_theory)
Concepts from linear algebra
AQ=[λ1𝐯1λ2𝐯2⋯λn𝐯n]. With this in mind, define the diagonal matrix Λ where each diagonal element Λii is the eigenvalue associated with the ith column of Q
Eigenvalues_and_eigenvectors
Type of geometric transformation
if S is a shear matrix with shear element λ, then Sn is a shear matrix whose shear element is simply nλ. Hence, raising a shear matrix to a power n multiplies
Shear_mapping
Algebraic element satisfying some of the criteria of an inverse
matrix A has no left inverse. X is a left inverse of A (XA=IU) X is a right inverse of A (AX=IV) The element b is a generalized inverse of an element
Generalized_inverse
Random matrix with gaussian entries
distribution. They are among the most-commonly studied matrix ensembles, fundamental to both mathematics and physics. The three main examples are the Gaussian orthogonal
Gaussian_ensemble
System for describing optical polarization
an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones
Jones_calculus
Physics model
the second spring slack. Gaussian network model Anisotropic Network Model Stiffness matrix Spring-mass system Laplacian matrix The Physics of Springs
Spring_system
Determinant of a subsection of a square matrix
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns
Minor_(linear_algebra)
Elements of a field, e.g. real numbers, in the context of linear algebra
Algebraic structure Scalar (physics) Linear algebra Matrix (mathematics) Row and column vectors Tensor Vector (mathematics and physics) Vector calculus Lay,
Scalar_(mathematics)
Algebraic object with geometric applications
high-dimensional matrix. Tensors have become important in physics, because they provide a concise mathematical framework for formulating and solving physics problems
Tensor
lepton mixing matrix U. Tribimaximal mixing is defined by a particular choice of the matrix of moduli-squared of the elements of the PMNS matrix as follows:
Tribimaximal_mixing
Partial order on matrices
functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering. Let A and B be two Hermitian
Loewner_order
Mathematical space
homogeneous n×k coordinate matrix W in [W] representing the element w∈𝐆𝐫k(V) does not affect the values of the affine coordinate matrix Ai1,…,ik representing
Grassmannian
Group of unitary complex matrices with determinant of 1
SU(3) group element generated by a traceless 3 × 3 Hermitian matrix H, normalized as tr(H2) = 2, can be expressed as a second order matrix polynomial in
Special_unitary_group
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Clifford algebra in 4 dimensions
in 1928 in developing the Dirac equation for spin-1/2 particles with a matrix representation of the gamma matrices, which represent the generators of
Dirac_algebra
Coordinates comprising a distance and two angles
dSφ=rdrdθ. The volume element spanning from r to r + dr, θ to θ + dθ, and φ to φ + dφ is specified by the determinant of the Jacobian matrix of partial derivatives
Spherical_coordinate_system
Mathematics concept
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose
Hadamard_matrix
Scientific subjects
physics, and molecular physics; optics and acoustics; condensed matter physics; high-energy particle physics and nuclear physics; and chaos theory and
Branches_of_physics
Transition rate formula
initial and final states of the system (described by the square of the matrix element of the perturbation) as well as the density of states. It is also applicable
Fermi's_golden_rule
Pulling force transmitted axially
purposes than tension. Stress is a 3x3 matrix called a tensor, and the σ 11 {\displaystyle \sigma _{11}} element of the stress tensor is tensile force
Tension_(physics)
Concept in differential geometry
the matrix with 1 at the (i, j)-th entry and zero on the other entries. The matrix Fij whose entries are 2-forms on P is called the curvature matrix. Given
Exterior_covariant_derivative
Mathematical operation in quantum optics, general relativity and other areas of physics
Gennes matrix. Also in nuclear physics, this method is applicable, since it may describe the "pairing energy" of nucleons in a heavy element. The Hilbert
Bogoliubov_transformation
Function acting on the space of physical states in physics
between bases. Each basis element ϕj can be connected to another, by the expression: Aij=⟨ϕi|Â|ϕj⟩, which is a matrix element: Â=(A11A12⋯A1nA21A22⋯A2n⋮⋮⋱⋮An1An2⋯Ann)
Operator_(physics)
Pictorial representation of the behavior of subatomic particles
particles or fields. The transition amplitude is then given as the matrix element of the S-matrix between the initial and final states of the quantum system.
Feynman_diagram
Soviet theoretical physicist (1908–1968)
in plasma physics, the Landau pole in quantum electrodynamics, the two-component theory of neutrinos, and Landau's equations for S-matrix singularities
Lev_Landau
Coordinate change in linear algebra
"new" (other) bases. A is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinates of the "new"
Change_of_basis
Formula in Lie group theory
exactly the same formula as in the matrix case. Left multiplication of an element of the algebra g by an element exp(X(t)) of the Lie group is interpreted
Derivative of the exponential map
Derivative_of_the_exponential_map
Generalization of gamma distribution to multiple dimensions
Dumitriu, Ioana; Edelman, Alan (2002). "Matrix models for beta ensembles". Journal of Mathematical Physics. 43 (11): 5830–5847. arXiv:math-ph/0206043
Wishart_distribution
Functions on special groups related to their matrix representations
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group
Matrix_coefficient
Symmetry transformation in particle physics for the strong and weak forces
In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group G, the elements of which do not obey
Non-abelian gauge transformation
Non-abelian_gauge_transformation
Group that is also a differentiable manifold with group operations that are smooth
widely used in many parts of modern mathematics and physics. Lie groups were first found by studying matrix subgroups G contained in GLn(ℝ) or GLn(ℂ), the
Lie_group
Computer graphics simulation of deformable objects
with scientific methods, particularly in the case of finite element simulations. Several physics engines currently provide software for soft-body simulation
Soft-body_dynamics
16-element matrix group
is a 16-element matrix group, consisting of the 4 Pauli matrices each with 4 possible phase factors. The n-qubit Pauli group is a 4n+1-element group consisting
Pauli_group
German physicist (1901–1976)
scattering matrix, or S-matrix, in elementary particle physics. The first two papers were published in 1943 and the third in 1944. The S-matrix described
Werner_Heisenberg
Constructions of Hadamard matrices
The Jacobsthal matrix has the properties QQT = qI − J and QJ = JQ = 0 where I is the q × q identity matrix and J is the q × q all 1 matrix. The first of
Paley_construction
Physical phenomenon
violation by modifying the matrix element to include vector and axial-vector couplings of fermions. This formed the matrix element that completed the Fermi
Beta_decay_transition
Type of group and algebra representation
group product is translated into matrix multiplication of the representations: D(ab)=D(a)D(b) If e is the identity element of the group (so that ae = ea
Irreducible_representation
Stochastic matrix representing links between entities
Assuming there are N pages, we can fill out A by doing the following: A matrix element A i , j {\displaystyle A_{i,j}} is filled with 1 if node j {\displaystyle
Google_matrix
Method for approximating eigenvalues
finite element method, the matrix Hkj is precisely the stiffness matrix of the Hamiltonian in the piecewise linear element space, and the matrix Skj is
Rayleigh–Ritz_method
Type of group in mathematics
of determinant 1, and therefore not an element of the component. By extension, for any field F, an n × n matrix with entries in F such that its inverse
Orthogonal_group
Requirement that quantum states' time evolution operators are unitary transformations
imaginary part of the S-matrix. In order to see what the right-hand side is, let us look at any specific element of this matrix, e.g. between some initial
Unitarity
&&\\&&&&&0&1\\&&&&&-1&0\end{array}}\right)~.} Each element of a circular ensemble is a unitary matrix, so it has eigenvalues on the unit circle: λ k = e
Circular_ensemble
Tool in linear algebra and matrix analysis
when element(s) of the output vector are zero. For example, when u or v is zero, we can eliminate the associated rows of the coefficient matrix without
Schur_complement
Algebra of 4D spacetime
Theoretical Physics. 43: 2061-2079. arXiv:math-ph/0405026. doi:10.1023/B:IJTP.0000049010.53558.b7. Sobczyk, Garret (2 November 2019). Matrix Gateway to
Algebra_of_physical_space
Two equalities that deal with the current and potential difference
current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German
Kirchhoff's_circuit_laws
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
Nonassociative algebra over the real numbers
vectors using a modified version of matrix multiplication. Specifically, define a vector-matrix to be a 2×2 matrix of the form [ a v w b ] , {\displaystyle
Split-octonion
Basis for the SU(3) Lie algebra
Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation
Gell-Mann_matrices
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix 𝐐 and a
Moment_of_inertia
Probability of a given process occurring in a particle collision
In physics, the cross section is a measure related to the probability that a specific process will take place in a collision of two particles. For example
Cross_section_(physics)
Idealization of a large number of atomic-sized systems
the observables to their expectation values. Density matrix – Mathematical tool in quantum physics Ensemble (fluid mechanics) – Imaginary collection of
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Chemical element with atomic number 80 (Hg)
chemical element; it has symbol Hg (from Latin hydrargyrum) and atomic number 80. It is commonly known as quicksilver. A heavy, silvery d-block element, mercury
Mercury_(element)
Type of physical quantity
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation
Pseudotensor
Chemical element with atomic number 112 (Cn)
stability In nuclear physics, an element is called heavy if its atomic number is high; lead (element 82) is one example of such a heavy element. The term "superheavy
Copernicium
Manner of referring to elements of arrays or tensors
different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication
Index_notation
Operation in mathematics
calculation purposes. That way, each row of the first matrix (1234), and each column of the second matrix (5678), point to the cell of the result that they
Tensor_contraction
Resistance to deformation in response to force
are M degrees of freedom a M×M matrix must be used to describe the stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses
Stiffness
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MATRIX ELEMENT-PHYSICS
MATRIX ELEMENT-PHYSICS
Boy/Male
English
Gentle. Famous Bearer: Clement Moore, writer of 'Twas the Night Before Christmas'.
Male
English
Pet form of English Matthew, MATTIE means "gift of God." Compare with feminine Mattie.
Surname or Lastname
English
English : patronymic from the personal name Clement.German, Dutch, and Danish : from the personal name Clemens (see Clement).Samuel Langhorne Clemens, better known by his pen name, Mark Twain, was descended from VA stock on his father’s side, from a Robert Clemens, who was born in Warwickshire, England, in 1634.
Male
English
 English form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Female
Welsh
Welsh form of Old French Caterine, CATRIN means "pure."
Female
German
Pet form of German Katarine, KATRIN means "pure."
Boy/Male
English American Danish
Gentle. Famous Bearer: Clement Moore, writer of 'Twas the Night Before Christmas'.
Male
English
Short form of Latin Clementius, CLEMENT means "gentle and merciful." meaning "gentle and merciful." In the bible, this is the name of a companion of Paul.
Male
English
Pet form of English Martin, MARTIE means "of/like Mars."
Boy/Male
English American Biblical Latin
Gentle. Famous Bearer: Clement Moore, writer of 'Twas the Night Before Christmas'.
Female
English
French form of Latin Maria, MARIE means "obstinacy, rebelliousness" or "their rebellion."
Male
French
 French form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Boy/Male
Australian, British, Danish, Dutch, English, Finnish, French, German, Irish, Latin, Swedish
Gentle; Merciful; Mild; Form of Clement
Female
English
English form of Latin Viatrix, BEATRIX means "voyager (through life)."
Female
Finnish
Finnish form of Greek Margarites, MAARIT means "pearl."
Female
English
Pet form of English Matilda, MATTIE means "mighty in battle." Compare with masculine Mattie.
Male
English
English surname transferred to forename use, derived from Latin Clemens or Clement, CLEMENTS means "gentle and merciful."
Surname or Lastname
English, French, and Dutch
English, French, and Dutch : from the Latin personal name Clemens meaning ‘merciful’ (genitive Clementis). This achieved popularity firstly through having been borne by an early saint who was a disciple of St. Paul, and later because it was selected as a symbolic name by a number of early popes. There has also been some confusion with the personal name Clemence (Latin Clementia, meaning ‘mercy’, an abstract noun derived from the adjective; in part a masculine name from Latin Clementius, a later derivative of Clemens). As an American family name, Clement has absorbed cognates in other continental European languages. (For forms, see Hanks and Hodges 1988.)
Female
Finnish
Finnish form of Greek Maria, MAARIA means "obstinacy, rebelliousness" or "their rebellion."Â
Girl/Female
Arabic, Australian, Basque, French, Latin
Lady; Feminine of Martin; Warlike
MATRIX ELEMENT-PHYSICS
MATRIX ELEMENT-PHYSICS
MATRIX ELEMENT-PHYSICS
MATRIX ELEMENT-PHYSICS
MATRIX ELEMENT-PHYSICS
MATRIX ELEMENT-PHYSICS
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