basic tensor

basic tensor

The present entry employs the terminology and notation defined and
described in the entry on

tensor arrays

. To keep things reasonably
self-contained we mention that the symbol

T

p

,

q

refers to the

vector space

of type

(

p

,

q

)

tensor arrays, i.e. maps

I

p

×

I

q

→

𝕂

,

where

I

is some finite list of
index labels, and where

𝕂

is a field.

We say that a tensor array is a

characteristic array

, a.k.a. a

basic tensor

, if all but one of its values are

0

, and the
remaining non-zero value is equal to

1

. For tuples

A

∈

I

p

and

B

∈

I

q

, we let

ε

A

B

:

I

p

×

I

q

→

𝕂

,

denote the characteristic
array defined by

(

ε

A

B

)

j

1

⁢

…

⁢

j

q

i

1

⁢

…

⁢

i

p

=

{

1

if

(

i

1

,

…

,

i

p

)

=

A

and

(

j

1

,

…

,

j

p

)

=

B

,

0

otherwise.

The type

(

p

,

q

)

characteristic arrays form a natural basis for

T

p

,

q

.

Furthermore the

outer multiplication

of two characteristic arrays
gives a characteristic array of larger valence. In other words, for

A

1

∈

I

p

1

,

B

1

∈

I

q

1

,

A

2

∈

I

p

2

,

B

2

∈

I

q

2

,

we have that

ε

A

1

B

1

⁢

ε

A

2

B

2

=

ε

A

1

⁢

A

2

B

1

⁢

B

2

,

where
the

product

on the left-hand side is performed by outer
multiplication, and where

A

1

⁢

A

2

on the right-hand side refers to
the element of

I

p

1

+

p

2

obtained by concatenating the tuples

A

1

and

A

2

, and similarly for

B

1

⁢

B

2

.

In this way we see that the type

(

1

,

0

)

characteristic arrays

ε

(

i

)

,

i

∈

I

(the natural basis of

𝕂

I

), and the type

(

0

,

1

)

characteristic arrays

ε

(

i

)

,

i

∈

I

(the natural basis of

(

𝕂

I

)

*

) generate the tensor array

algebra

relative to the
outer multiplication

operation

.

The just-mentioned fact gives us an alternate way of writing and
thinking about tensor arrays. We introduce the basic symbols

ε

(

i

)

,

ε

(

i

)

,

i

∈

I

subject to the commutation

relations

ε

(

i

)

⁢

ε

(

i

′

)

=

ε

(

i

′

)

⁢

ε

(

i

)

,

i

,

i

′

∈

I

,

add and multiply these symbols using coefficients in

𝕂

, and use

ε

(

j

1

⁢

…

⁢

j

p

)

(

i

1

⁢

…

⁢

i

q

)

,

i

1

,

…

,

i

q

,

j

1

,

…

,

j

p

∈

I

as a handy abbreviation for

ε

(

i

1

)

⁢

…

⁢

ε

(

i

q

)

⁢

ε

(

j

1

)

⁢

…

⁢

ε

(

j

p

)

.

We
then interpret the resulting

expressions

as tensor arrays in the

obvious

fashion: the values of the tensor array are just the
coefficients of the

ε

symbol matching the given index. However,
note that in the

ε

symbols, the covariant data is written as a

superscript

, and the contravariant data as a

subscript

. This is done
to facilitate the

Einstein summation convention

.

By way of illustration, suppose that

I

=

(

1

,

2

)

. We can now write down
a type

(

1

,

0

)

tensor, i.e. a

column vector

u

=

(

u

1

u

2

)

∈

T

1

,

0

as

u

=

u

1

⁢

ε

(

1

)

+

u

2

⁢

ε

(

2

)

.

Similarly, a row-vector

ϕ

=

(

ϕ

1

,

ϕ

2

)

∈

T

0

,

1

can be written down as

ϕ

=

ϕ

1

⁢

ε

(

1

)

+

ϕ

2

⁢

ε

(

2

)

.

In the case of a matrix

M

=

(

M

1

1

M

1

2

M

2

1

M

2

2

)

∈

T

1

,

1

we would write

M

=

M

1

1

⁢

ε

(

1

)

(

1

)

+

M

2

1

⁢

ε

(

1

)

(

2

)

+

M

1

2

⁢

ε

(

2

)

(

1

)

+

M

2

2

⁢

ε

(

2

)

(

2

)

.

Title

basic tensor

Canonical

name

BasicTensor

Date of creation

2013-03-22 12:40:37

Last modified on

2013-03-22 12:40:37

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Derivation

Classification

msc 15A69

Synonym

characteristic array

Related topic

TensorArray

Related topic

Basis

Related topic

Frame

Related topic

SimpleTensor

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