tensor transformations

tensor transformations

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

tensor arrays

and

basic tensors

. 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. The symbols

ε

(

i

)

,

ε

(

i

)

,

i

∈

I

refer to the column and

row vectors

giving the
natural basis of

T

1

,

0

and

T

0

,

1

, respectively.

Let

I

and

J

be two finite lists of equal

cardinality

, and let

T

:

𝕂

I

→

𝕂

J

be a

linear isomorphism

. Every
such

isomorphism

is uniquely represented by an

invertible matrix

M

:

J

×

I

→

𝕂

with entries given by

M

i

j

=

(

T

⁢

ε

(

i

)

)

j

,

i

∈

I

,

j

∈

J

.

In other words, the
action of

T

is described by the following substitutions

ε

(

i

)

↦

∑

j

∈

J

M

i

j

⁢

ε

(

j

)

,

i

∈

I

.

(1)

Equivalently, the action of

T

is given by matrix-multiplication
of

column vectors

in

𝕂

I

by

M

.

The corresponding substitutions

relations

for the type

(

0

,

1

)

tensors
involve the inverse matrix

M

-

1

:

I

×

J

→

𝕂

,

and
take the form

1

1

The above relations describe the action of the

dual homomorphism

of the

inverse

transformation

(

T

-

1

)

*

:

(

𝕂

I

)

*

→

(

𝕂

J

)

*

.

ε

(

i

)

↦

∑

j

∈

J

(

M

-

1

)

j

i

⁢

ε

(

j

)

,

i

∈

I

.

(2)

The rules for type

(

0

,

1

)

substitutions are what they are, because of
the requirement that the

ε

(

i

)

and

ε

(

i

)

remain dual bases
even after the substitution. In other words we want the substitutions
to preserve the relations

ε

(

i

1

)

⁢

ε

(

i

2

)

=

δ

i

2

i

1

,

i

1

,

i

2

∈

I

,

where the left-hand side of the above

equation

features the

inner
product

and the right-hand side the

Kronecker delta

. Given that the
vector basis transforms as in (

1

) and given the above
constraint, the substitution rules for the

linear form

basis, shown in
(

2

), are the only such possible.

The classical terminology of contravariant and covariant indices is
motivated by thinking in term of substitutions. Thus, suppose we
perform a linear substitution and change a vector, i.e. a type

(

1

,

0

)

tensor,

X

∈

𝕂

I

into a vector

Y

∈

𝕂

J

. The indexed
values of the former and of the latter are related by

Y

j

=

∑

i

∈

I

M

i

j

⁢

X

i

,

j

∈

J

.

(3)

Thus, we see
that the “transformation rule” for indices is contravariant to the
substitution rule (

1

) for basis vectors.

In modern terms, this contravariance is best described by saying that
the

dual space

space construction is a

contravariant functor

2

2

See the entry on the dual homomorphism.

. In other
words, the substitution rule for the linear forms, i.e. the type

(

0

,

1

)

tensors, is contravariant to the substitution rule for
vectors:

ε

(

j

)

↦

∑

i

∈

I

M

i

j

⁢

ε

(

i

)

,

j

∈

J

,

(4)

in full agreement with the relation shown in (

2

).
Everything comes together, and equations (

3

) and
(

4

) are seen to be one and the same, once we remark
that tensor array values can be obtained by contracting with
characteristic arrays. For example,

X

i

=

ε

(

i

)

⁢

(

X

)

,

i

∈

I

;

Y

j

=

ε

(

j

)

⁢

(

Y

)

,

j

∈

J

.

Finally we must remark that the transformation rule for covariant
indices involves the inverse matrix

M

-

1

. Thus if

α

∈

T

0

,

1

⁢

(

I

)

is transformed to a

β

∈

T

0

,

1

the
indices will be related by

β

j

=

∑

i

∈

I

(

M

-

1

)

j

i

⁢

α

i

,

j

∈

J

.

The most general transformation rule for tensor array indices is
therefore the following: the indexed values of a tensor array

X

∈

T

p

,

q

⁢

(

I

)

and the values of the transformed tensor array

Y

∈

T

p

,

q

⁢

(

J

)

are related by

Y

l

1

⁢

…

⁢

l

q

j

1

⁢

…

⁢

j

p

=

∑

i

1

,

…

,

i

p

∈

I

p

k

1

,

…

,

k

q

∈

I

q

M

i

1

j

1

⁢

⋯

⁢

M

i

p

j

p

⁢

(

M

-

1

)

l

1

k

1

⁢

⋯

⁢

(

M

-

1

)

l

q

k

q

⁢

X

k

1

⁢

…

⁢

k

q

i

1

⁢

…

⁢

i

p

,

for all possible choice of indices

j

1

,

…

⁢

j

p

,

l

1

,

…

,

l

q

∈

J

.

Debauche of indices, indeed!

Title

tensor transformations

Canonical name

TensorTransformations

Date of creation

2013-03-22 12:40:33

Last modified on

2013-03-22 12:40:33

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

4

Author

rmilson (146)

Entry type

Derivation

Classification

msc 15A69

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