# tensor array

## Introduction.

, or tensors for short^{1}^{1}The 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 $\mathrm{\pi \x9d\x95\x82}$ be a field^{2}^{2}In physics and differential
geometry, $\mathrm{\pi \x9d\x95\x82}$ is typically $\mathrm{\beta \x84\x9d}$ or $\mathrm{\beta \x84\x82}$. and let $I$
be a finite list of indices^{3}^{3}It 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 $(1,2,\mathrm{\beta \x80\xa6},n)$. A tensor array of type

$$(p,q),p,q\beta \x88\x88\mathrm{\beta \x84\x95}$$ |

is a mapping

$${I}^{p}\Gamma \x97{I}^{q}\beta \x86\x92\mathrm{\pi \x9d\x95\x82}.$$ |

The set of all such mappings will be denoted by ${\mathrm{T}}^{p,q}\beta \x81\u2019(I,\mathrm{\pi \x9d\x95\x82})$, or when $I$ and $\mathrm{\pi \x9d\x95\x82}$ 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 $\mathrm{\pi \x9d\x95\x82}$. It is also customary to
identify ${\mathrm{T}}^{1,0}$ with ${\mathrm{\pi \x9d\x95\x82}}^{I}$, the vector space of list
vectors indexed by $I$, and to identify ${\mathrm{T}}^{0,1}$ with dual space^{}
${\left({\mathrm{\pi \x9d\x95\x82}}^{I}\right)}^{*}$ of linear forms^{} on ${\mathrm{\pi \x9d\x95\x82}}^{I}$. Finally,
${\mathrm{T}}^{0,0}$ can be identified with $\mathrm{\pi \x9d\x95\x82}$ itself. In other
words, scalars are tensor arrays of zero valence.

Let $X:{I}^{p}\Gamma \x97{I}^{q}\beta \x86\x92\mathrm{\pi \x9d\x95\x82}$ 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 ${i}_{1},\mathrm{\beta \x80\xa6},{i}_{p},{j}_{1},\mathrm{\beta \x80\xa6},{j}_{q}\beta \x88\x88I$ we write

$${X}_{{j}_{1}\beta \x81\u2019\mathrm{\beta \x80\xa6}\beta \x81\u2019{j}_{q}}^{{i}_{1}\beta \x81\u2019\mathrm{\beta \x80\xa6}\beta \x81\u2019{i}_{p}}$$ |

instead of ^{4}^{4}Curiously, the latter notation is preferred by some
authors. See H. Weylβs books and papers, for example.

$$X\beta \x81\u2019({i}_{1},\mathrm{\beta \x80\xa6},{i}_{p};{j}_{1},\mathrm{\beta \x80\xa6},{j}_{p}).$$ |

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.^{5}^{5}It 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 doesnβt 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 |