tensor array

Introduction.

, or tensors for short¹¹The term tensor has other meanings, c.f. the tensor entry (http://planetmath.org/Tensor). are multidimensional arrays with two types of (covariant and contravariant) indices. Tensors are widely used in science and mathematics, because these data structures are the natural choice of representation for a variety of important physical and geometric quantities.

In this entry we give the definition of a tensor array and establish some related terminology and notation. The theory of tensor arrays incorporates a number of other essential topics: basic tensors, tensor transformations, outer multiplication, contraction, inner multiplication, and generalized transposition. These are fully described in their separate entries.

Valences and the space of tensors arrays.

Let $\mathbb{K}$ be a field²²In physics and differential geometry, $\mathbb{K}$ is typically $\mathbb{R}$ or $\mathbb{C}$ . and let $I$ be a finite list of indices³³It is advantageous to allow general indexing sets, because one can indicate the use of multiple frames of reference by employing multiple, disjoint sets of indices., such as $(1,2,\ldots,n)$ . A tensor array of type

$(p,q),\quad p,q\in\mathbb{N}$

is a mapping

$I^{p}\times I^{q}\rightarrow\mathbb{K}.$

The set of all such mappings will be denoted by $\mathrm{T}^{p,q}(I,\mathbb{K})$ , or when $I$ and $\mathbb{K}$ are clear from the context, simply as $\mathrm{T}^{p,q}$ . The numbers $p$ and $q$ are called, respectively, the contravariant and the covariant valence of the tensor array.

Point-wise addition and scaling give $\mathrm{T}^{p,q}$ the structure of a a vector space of dimension $n^{p+q}$ , where $n$ is the cardinality of $I$ . We will interpret $I^{0}$ as signifying a singleton set. Consequently $\mathrm{T}^{p,0}$ and $\mathrm{T}^{0,q}$ are just the maps from, respectively, $I^{p}$ and $I^{q}$ to $\mathbb{K}$ . It is also customary to identify $\mathrm{T}^{1,0}$ with $\mathbb{K}^{I}$ , the vector space of list vectors indexed by $I$ , and to identify $\mathrm{T}^{0,1}$ with dual space $\left(\mathbb{K}^{I}\right)^{*}$ of linear forms on $\mathbb{K}^{I}$ . Finally, $\mathrm{T}^{0,0}$ can be identified with $\mathbb{K}$ itself. In other words, scalars are tensor arrays of zero valence.

Let $X:I^{p}\times I^{q}\rightarrow\mathbb{K}$ be a type $(p,q)$ tensor array. In writing the values of $X$ , it is customary to write contravariant indices using superscripts, and covariant indices using subscripts. Thus, for indices $i_{1},\ldots,i_{p},j_{1},\ldots,j_{q}\in I$ we write

$X^{i_{1}\ldots i_{p}}_{j_{1}\ldots j_{q}}$

instead of ⁴⁴Curiously, the latter notation is preferred by some authors. See H. Weyl’s books and papers, for example.

$X(i_{1},\ldots,i_{p};j_{1},\ldots,j_{p}).$

We also mention that it is customary to use columns to represent contravariant index dimensions, and rows to represent the covariant index dimensions. Thus column vectors are type $(1,0)$ tensor arrays, row vectors are type $(0,1)$ tensor arrays, and matrices, in as much as they can be regarded either as rows of columns or as columns of rows, are type $(1,1)$ tensor arrays.⁵⁵It is also customary to use matrices to also represent type $(2,0)$ and type $(0,2)$ tensor arrays (The latter are used to represent quadratic forms.) Speaking idealistically, such objects should be typeset, respectively, as a column of column vectors and as a row of row vectors. However typographical constraints and notational convenience dictate that they be displayed as matrices.

Notes.

It must be noted that our usage of the term tensor array is non-standard. The traditionally inclined authors simply call these data structures tensors. We bother to make the distinction because the traditional nomenclature is ambiguous and doesn’t include the modern mathematical understanding of the tensor concept. (This is explained more fully in the tensor entry (http://planetmath.org/Tensor).) Precise and meaningful definitions can only be given by treating the concept of a tensor array as distinct from the concept of a geometric/abstract tensor.

We also mention that the term tensor is often applied to objects that should more appropriately be termed a tensor field. The latter are tensor-valued functions, or more generally sections of a tensor bundle. A tensor is what one gets by evaluating a tensor field at one point. Informally, one can also think of a tensor field as a tensor whose values are functions, rather than constants.

Title	tensor array
Canonical name	TensorArray
Date of creation	2013-03-22 12:40:25
Last modified on	2013-03-22 12:40:25
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	10
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A69
Related topic	Frame
Related topic	Vector2
Related topic	BasicTensor
Related topic	TensorProductClassical
Related topic	Tensor
Defines	covariant index
Defines	contravariant index
Defines	valence