tensor array

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

𝕂

be a field

2

2

In physics and differential
geometry,

𝕂

is typically

ℝ

or

ℂ

.

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

,

…

,

n

)

. A tensor array of type

(

p

,

q

)

,

p

,

q

∈

ℕ

is a

mapping

I

p

×

I

q

→

𝕂

.

The set of all such mappings
will be denoted by

T

p

,

q

⁢

(

I

,

𝕂

)

, or when

I

and

𝕂

are clear from the

context

, simply as

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

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

T

p

,

0

and

T

0

,

q

are just the maps from,
respectively,

I

p

and

I

q

to

𝕂

. It is also customary to
identify

T

1

,

0

with

𝕂

I

, the vector space of

list
vectors

indexed by

I

, and to identify

T

0

,

1

with

dual space

(

𝕂

I

)

*

of

linear forms

on

𝕂

I

. Finally,

T

0

,

0

can be identified with

𝕂

itself. In other
words, scalars are tensor arrays of zero valence.

Let

X

:

I

p

×

I

q

→

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

,

…

,

i

p

,

j

1

,

…

,

j

q

∈

I

we write

X

j

1

⁢

…

⁢

j

q

i

1

⁢

…

⁢

i

p

instead of

4

4

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

X

⁢

(

i

1

,

…

,

i

p

;

j

1

,

…

,

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 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

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