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Graphical notation for multilinear algebra calculations
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions
Penrose_graphical_notation
Algebraic object with geometric applications
distinct pairs of indices may be summed this way. Penrose graphical notation is a diagrammatic notation which replaces the symbols for tensors with shapes
Tensor
British geneticist
Shirley Victoria Penrose Hodgson (born 22 February 1945) is a British geneticist. Hodgson studied at Somerville College, Oxford. She worked as a GP, then
Shirley_Hodgson
Graphical representation of a morphism
tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics
String_diagram
Mathematical study of illumination of rooms with mirrored walls
The original problem was first solved in 1958 by Roger Penrose using ellipses to form the Penrose unilluminable room. He showed that there exists a room
Illumination_problem
Mathematical notation for tensors and spinors
_{3}}\omega _{\sigma (a)\sigma (b)\sigma (c)}} Penrose graphical notation Einstein notation Index notation Tensor Antisymmetric tensor Raising and lowering
Abstract_index_notation
System of symbolic representation
mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are Penrose graphical notation and Coxeter–Dynkin
Mathematical_notation
Model of quantum computing
physical cables. The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation.[citation needed] Richard
Quantum_circuit
Topics referred to by the same term
notation (dance) A diagrammatic notation in mathematical notation In physics: Penrose graphical notation Coxeter–Dynkin diagram A visual programming language
Graphic_notation
Roger Penrose: Moore–Penrose inverse, the most widely known generalization of the inverse matrix in particular linear algebra Penrose graphical notation, a
List of things named after Roger Penrose
List_of_things_named_after_Roger_Penrose
Graphical language for quantum processes
These are connected together to form a tensor network similar to Penrose graphical notation. Due to the symmetries of the spiders and the properties of the
ZX-calculus
British theoretical physicist (born 1929)
Oliver Penrose FRS FRSE (born 6 June 1929) is a British theoretical physicist and emeritus professor at Heriot-Watt University. His topics of interest
Oliver_Penrose
Mathematical Concept
associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas
Voigt_notation
Shorthand notation for tensor operations
\alpha }} Tensor Abstract index notation Bra–ket notation Penrose graphical notation Levi-Civita symbol DeWitt notation This applies only for numerical
Einstein_notation
Claim that human mathematicians are not describable as formal proof systems
The Penrose–Lucas argument is a logical argument partially based on Kurt Gödel's first incompleteness theorem. In 1931, Gödel proved that every effectively
Penrose–Lucas_argument
Diagram used to represent quantum field theory calculations
representations of matrix groups. The diagrammatic notation can thus greatly simplify calculations. Roger Penrose described spin networks in 1971. Spin networks
Spin_network
Tensor index notation for tensor-based calculations
Metric tensor Multilinear algebra Multilinear subspace learning Penrose graphical notation Regge calculus Ricci decomposition Tensor (intrinsic definition)
Ricci_calculus
Origin and evolution of the symbols used to write equations and formulas
fields called a tetrad). In the 1990s, Roger Penrose proposed Penrose graphical notation (tensor diagram notation) as a, usually handwritten, visual depiction
History of mathematical notation
History_of_mathematical_notation
Philosophical argument based on the theory of relativity
Putnam (1967). It is sometimes called the Rietdijk–Putnam–Penrose argument. Roger Penrose advanced a form of this argument that has been called the Andromeda
Rietdijk–Putnam_argument
Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory
Multi-index_notation
Branch of mathematics
tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning Multivector Geometric algebra Clifford algebra
Multilinear_algebra
English mathematician, mathematical physicist (born 1931)
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, and philosopher of science. He is Emeritus Rouse Ball Professor
Roger_Penrose
Algebraic operation on coordinate vectors
specified with respect to an orthonormal basis, is defined, in summation notation, as: a ⋅ b = ∑ i = 1 n a i b i = a 1 b 1 + a 2 b 2 + ⋯ + a n b n {\displaystyle
Dot_product
contrast, a dyad is specifically a dyadic tensor of rank one. Einstein notation This notation is based on the understanding that whenever a multidimensional array
Glossary_of_tensor_theory
Tensor equal to the negative of any of its transpositions
Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7. Penrose, Roger (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4
Antisymmetric_tensor
Quantum state of multiple particles represented as complex matrices
and. In the context of finite automata see. For emphasis placed on the graphical reasoning of tensor networks, see the introduction. For a system of N
Matrix_product_state
Pictorial computational technique in quantum chemistry
notation and include the abstract nature of the state, such as tensor products and transformation rules. The notation parallels the idea of Penrose graphical
Angular momentum diagrams (quantum mechanics)
Angular_momentum_diagrams_(quantum_mechanics)
Topological space that locally resembles Euclidean space
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Manifold
Coordinate-free definition of a tensor
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Tensor_(intrinsic_definition)
Specialized notation for multivariable calculus
calculus. Supports general symbolic tensor derivatives using Penrose graphical notation. Matrix Reference Manual, Mike Brookes, Imperial College London
Matrix_calculus
Notation used for Weyl spinors
In theoretical physics, Van der Waerden notation refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard
Van_der_Waerden_notation
Effect in special relativity
also sometimes referred to as the Penrose–Terrell effect, the Terrell–Penrose effect or the Lampa–Terrell–Penrose effect. In 1924, a paper by Anton Lampa
Terrell_rotation
Graphical means of performing computations in linear algebra
have simple diagrammatic proofs. They are closely related to Penrose's graphical notation. Let V be a vector space of dimension n over a field F (with
Trace_diagram
Isomorphism between the tangent and cotangent bundles of a manifold
the use of the musical notation symbols ♭ {\displaystyle \flat } (flat) and ♯ {\displaystyle \sharp } (sharp). In the notation of Ricci calculus and mathematical
Musical_isomorphism
Matrix operation which flips a matrix over its diagonal
another matrix, called the transpose of A and often denoted AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician
Transpose
Function that is invariant under all permutations of its variables
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Symmetric_function
Array of numbers
or no columns, called an empty matrix. The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written
Matrix_(mathematics)
Mathematical operation on vector spaces
differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may
Tensor_product
Specification of a derivative along a tangent vector of a manifold
language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension
Covariant_derivative
Tensor invariant under permutations of vectors it acts on
the operator is omitted: T1T2 = T1 ⊙ T2. In some cases an exponential notation is used: v ⊙ k = v ⊙ v ⊙ ⋯ ⊙ v ⏟ k times = v ⊗ v ⊗ ⋯ ⊗ v ⏟ k times =
Symmetric_tensor
Differential form of degree one or section of a cotangent bundle
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
One-form
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
developing Aitken's diagrams, to become part of the technique of Penrose graphical notation. Also, this relation is extensively used in S-duality theories
Kronecker_delta
Mapping from p forms to p-1 forms
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Interior_product
Physics concept
{x}^{i}}}} This is the explicit form of the covariant transformation rule. The notation of a normal derivative with respect to the coordinates sometimes uses a
Covariant_transformation
Conserved physical quantity; rotational analogue of linear momentum
about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin
Angular_momentum
Theory of gravitation as curved spacetime
introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969 Bekenstein 1973, Bekenstein 1974 The fact that black holes
General_relativity
Tensor having both covariant and contravariant indices
covariant, the last one contravariant, and the remaining ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance
Mixed_tensor
Mathematical function, in linear algebra
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Linear_map
Belgian theoretical physicist and logician
the development of a diagrammatic quantum formalism based on Penrose graphical notation, on which he wrote a textbook entitled Picturing Quantum Processes
Bob_Coecke
Assignment of a tensor continuously varying across a region of space
transforms by the inverse Jacobian. The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent
Tensor_field
Tensor that describes the 4D geometry of spacetime
{\displaystyle g_{\mu \nu }} themselves as the metric (see, however, abstract index notation). With the quantities d x μ {\displaystyle dx^{\mu }} being regarded as
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Measure of the curvature of a pseudo-Riemannian manifold
v_{3}\right)k\left(v_{1},v_{4}\right)\end{aligned}}} In tensor component notation, this can be written as C i k ℓ m = R i k ℓ m + 1 n − 2 ( R i m g k ℓ −
Weyl_tensor
Abbreviation in the fields of special and general relativity
four-dimensional spacetime. General four-tensors are usually written in tensor index notation as A ν 1 , ν 2 , . . . , ν m μ 1 , μ 2 , . . . , μ n {\displaystyle A_{\;\nu
Four-tensor
Continuous surjection satisfying a local triviality condition
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Fiber_bundle
Algebra associated to any vector space
algebra, a quantum deformation of the symmetric algebra by a symplectic form Penrose, R. (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4
Exterior_algebra
Expression that may be integrated over a region
dependent is zero. A common notation for the wedge product of elementary k {\displaystyle k} -forms is so called multi-index notation: in an n {\displaystyle
Differential_form
Affine connection on the tangent bundle of a manifold
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Levi-Civita_connection
British physiologist and biochemist (1864–1956)
Leathes married Lionel Penrose MD, FRS, Professor of Genetics at University College, London in 1928, and was the mother of Oliver Penrose, the theoretical physicist;
John_Beresford_Leathes
Concept in differential geometry
form ψ is horizontal if ψ(v0, ..., vk) = ψ(hv0, ..., hvk).) By abuse of notation, the differential of ρ at the identity element may again be denoted by
Exterior_covariant_derivative
Tensor in differential geometry
v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation, R i c a b = R c b c a = R c a c b . {\displaystyle \mathrm {Ric} _{ab}=\mathrm
Ricci_curvature
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
\end{aligned}}} It is common in rigid body mechanics to use notation that explicitly identifies the x {\displaystyle x} , y {\displaystyle y}
Moment_of_inertia
Operation in mathematics
2x2; often 3x3 or 4x4 are used, but any size is allowed. In simple index notation, this is written ∑ j = 1 2 a i j × b j k = c i k {\textstyle \sum _{j=1}^{2}a_{ij}\times
Tensor_contraction
Electromagnetism in general relativity
square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is F α β ; γ =
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Type of physical quantity
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Pseudotensor
Decomposition in multilinear algebra
{\displaystyle M>2} and all I m ≥ 2 {\displaystyle I_{m}\geq 2} . For simplicity in notation, assume without loss of generality that the factors are ordered such that
Tensor_rank_decomposition
Operation that pairs a left and a right R-module into an abelian group
_{R}N} . It is often called a pure tensor. Strictly speaking, the correct notation would be x ⊗R y but it is conventional to drop R here. Then, immediately
Tensor_product_of_modules
Method for specifying point positions
coordinates Frame of reference Galilean transformation Grid reference Nomogram, graphical representations of different coordinate systems Reference system Rotation
Coordinate_system
Differential form
{\displaystyle \omega } is frequently used to denote the volume form, this notation is not universal; the symbol ω {\displaystyle \omega } often carries many
Volume_form
Graph-structured representation of high-dimensional tensors and quantum many-body states
formalism is its diagrammatic calculus, descending from the graphical notation introduced by Roger Penrose, in which multilinear-algebraic manipulations are performed
Tensor_network
Study of curves from a differential point of view
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Differentiable_curve
Basis used to express spherical tensors
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Spherical_basis
Property of a mathematical space
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Dimension
Quantum mechanics posed in terms of category theory
diagrams. These diagrammatic languages can be traced back to Penrose graphical notation, developed in the early 1970s. Diagrammatic reasoning has been
Categorical_quantum_mechanics
Vector behavior under coordinate changes
opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm. In Einstein notation, contravariant vectors and components of tensors are shown with superscripts
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Antisymmetric permutation object acting on tensors
lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:
Levi-Civita_symbol
Set of vectors used to define coordinates
j}y_{j},} for i = 1, ..., n. This formula may be concisely written in matrix notation. Let A be the matrix of the a i , j {\displaystyle a_{i,j}} , and X = [
Basis_(linear_algebra)
Straight path on a curved surface or a Riemannian manifold
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Geodesic
Tensor describing energy momentum density in spacetime
superscripted variables (not exponents; see Tensor index notation and Einstein summation notation). The four coordinates of an event of spacetime x are given
Stress–energy_tensor
Tensor field in Riemannian geometry
noncommutativity of the second covariant derivative. In abstract index notation, R d c a b Z c = ∇ a ∇ b Z d − ∇ b ∇ a Z d . {\displaystyle R^{d}{}_{cab}Z^{c}=\nabla
Riemann_curvature_tensor
Mathematical object that describes the electromagnetic field in spacetime
}F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0} or using the index notation with square brackets[note 1] for the antisymmetric part of the tensor:
Electromagnetic_tensor
Exterior algebraic map taking tensors from p forms to n-p forms
}(dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}} These are summarized in the index notation as ⋆ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ , ⋆ ( d x
Hodge_star_operator
Non-tensorial representation of the spin group
in Mathematics. 14: 1–55. doi:10.1016/0001-8708(74)90021-8. MR 0358873. Penrose, Roger; Rindler, W. (1988). Spinor and twistor methods in space-time geometry
Spinor
Spinning motion in theoretical physics
{\mathfrak {se}}(d)} . This article uses Cartesian coordinates and tensor index notation. The Noether current for translations in space is momentum, while the current
Spin_tensor
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Symmetrization
Representation of a tensor in Euclidean space
ayey (a vector), and similarly for x and z. A more general notation is tensor index notation, which has the flexibility of numerical values rather than
Cartesian_tensor
Math/physics concept
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Connection_form
Object in differential geometry
part and another part which contains the trace terms. Using the index notation, the trace of T is given by a i = T k i k , {\displaystyle a_{i}=T^{k}{}_{ik}
Torsion_tensor
General relativity articles using tensors will use the abstract index notation. The principle of general covariance was one of the central principles
Mathematics of general relativity
Mathematics_of_general_relativity
Universal construction in multilinear algebra
was actually one and the same thing as ∇ {\displaystyle \nabla } ; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor
Tensor_algebra
Physical phenomenon
Bibcode:2010ConPh..51...59C. doi:10.1080/00107510903257624. S2CID 752173. R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics
Quantum_teleportation
Theory of interwoven space and time by Albert Einstein
would be observed as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals
Special_relativity
Generalization of tensor fields
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Tensor_density
Tensor operator generalizes the notion of operators which are scalars and vectors
Italiana di Fisica, IOS. ISBN 978-905-199-24-72. Introduction to the Graphical Theory of Angular Momentum. Springer. 2009. ISBN 978-364-203-11-99. A
Tensor_operator
Structure defining distance on a manifold
is increased by du units, and v is increased by dv units. Using matrix notation, the first fundamental form becomes d s 2 = [ d u d v ] [ E F F G ] [ d
Metric_tensor
Construct allowing differentiation of tangent vector fields of manifolds
Y]=\left(X^{j}\partial _{j}Y^{i}-Y^{j}\partial _{j}X^{i}\right)\partial _{i}} in Einstein notation. This is independent of coordinate system choice and ∂ i = ( ∂ ∂ ξ i )
Affine_connection
Array of numbers describing a metric connection
reminder that these are defined to be equivalent notation for the same concept. The choice of notation is according to style and taste, and varies from
Christoffel_symbols
Branch of physics which studies the behavior of materials modeled as continuous media
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Continuum_mechanics
System of moving vectors in differential geometry
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Parallel_transport
Geometric structure
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Spinor_bundle
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