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  2. Tensor - Wikipedia

    en.wikipedia.org/wiki/Tensor

    A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.

  3. Tensor (machine learning) - Wikipedia

    en.wikipedia.org/wiki/Tensor_(machine_learning)

    In machine learning, the term tensor informally refers to two different concepts for organizing and representing data. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector space.

  4. String diagram - Wikipedia

    en.wikipedia.org/wiki/String_diagram

    When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.

  5. Glossary of tensor theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_tensor_theory

    The earliest foundation of tensor theory – tensor index notation. [1] Order of a tensor The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank. Rank of a tensor The rank of a tensor is the minimum ...

  6. Tensor (intrinsic definition) - Wikipedia

    en.wikipedia.org/wiki/Tensor_(intrinsic_definition)

    A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor [1]) is a tensor that can be written as a product of tensors of the form = where a, b, ..., d are nonzero and in V or V ∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors.

  7. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays. A tensor may be expressed as a linear sum of the tensor product of vector and covector basis ...

  8. Abstract index notation - Wikipedia

    en.wikipedia.org/wiki/Abstract_index_notation

    Abstract index notation handles braiding as follows. On a particular tensor product, an ordering of the abstract indices is fixed (usually this is a lexicographic ordering). The braid is then represented in notation by permuting the labels of the indices. Thus, for instance, with the Riemann tensor

  9. Cartesian tensor - Wikipedia

    en.wikipedia.org/wiki/Cartesian_tensor

    A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors e x ↦ e 1, e y ↦ e 2, e z ↦ e 3 and coordinates a x ↦ a 1, a y ↦ a 2, a z ↦ a 3.