Everything about Tensor
Algebraic object with geometric applications
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors, dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: TensorA 5-minute orientation
- Skim a recent paper: The ITensor Software Library for Tensor Network CalculationsInfluential work since 2022
- Pick up a classic book: Schaum's outline of theory and problems of vector analysisThe most-reprinted book on the subject
- Read the landmark paper: Tensor Decompositions and ApplicationsCited 8,728 times
- Go sideways: Covariance and contravariance of vectorsA closely related topic
- Go sideways: Tensor contractionA closely related topic
📖 Definitions
Noun A muscle that tightens or stretches a part, or renders it tense.
Noun A mathematical object that describes linear relations on scalars, vectors, matrices and other algebraic objects, and is represented as a multidimensional array.
Noun A multidimensional array with (at least) two dimensions.
Noun A norm operation on the quaternion algebra.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Tensor Decompositions and Applications
MR diffusion tensor spectroscopy and imaging
Subgrid-Scale Stress Modelling Based on the Square of the Velocity Gradient Tensor
How to correct susceptibility distortions in spin-echo echo-planar images: application to diffusion tensor imaging
Microstructural and Physiological Features of Tissues Elucidated by Quantitative-Diffusion-Tensor MRI
Estimation of the Effective Self-Diffusion Tensor from the NMR Spin Echo
Diffusion tensor imaging: Concepts and applications
The Einstein Tensor and Its Generalizations
🆕 Influential since 2022
The ITensor Software Library for Tensor Network Calculations
Diffusion Tensor Image Analysis ALong the Perivascular Space (DTI-ALPS): Revisiting the Meaning and Significance of the Method
Hyperspectral Image Denoising via Tensor Low-Rank Prior and Unsupervised Deep Spatial–Spectral Prior
Integrated Sensing and Communication With Massive MIMO: A Unified Tensor Approach for Channel and Target Parameter Estimation
Logic Tensor Networks
📚 Books worth reading
Schaum's outline of theory and problems of vector analysisMurray R. Spiegel · 1959
Physical properties of crystals, their representation by tensors and matricesJ. F. Nye · 1957
TensoranalysisHeinz Schade · 1996
Tensor calculusBarry Spain · 1953
Matrices and tensors in physicsA. W. Joshi · 1975
Tensor calculusBarry Spain · 1956
Tensor calculusJ. L. Synge · 1949
Vector calculus, linear algebra, and differential formsHubbard, John H., John H. Hubbard · 1999
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Covariance and contravariance of vectors In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric…
- Tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. This example…
- Tensor (intrinsic definition) In mathematics, the modern component-free approach to the theory of a tensor views it as an abstract object, expressing some definite type of multilinear…
- Glossary of tensor theory This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application…
- Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with…
- Tensor density In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor…
- Covariant transformation In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transforma…
- Einstein notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein…