Tensor on a vector space
over a field
An element of the vector space
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where is the dual space of
. The tensor
is said to be
times contravariant and
times covariant, or to be of type
. The number
is called the contravariant valency, and
the covariant valency, while the number
is called the general valency of the tensor
. The space
is identified with
. Tensors of type
are called contravariant, those of the type
are called covariant, and the remaining ones are called mixed.
Examples of tensors.
1) A vector of the space (a tensor of type
).
2) A covector of the space (a tensor of type
).
3) Any covariant tensor
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where , defines a
-linear form
on
by the formula
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the mapping from the space
into the space
of all
-linear forms on
is linear and injective; if
, then this mapping is an isomorphism, since any
-linear form corresponds to some tensor of type
.
4) Similarly, a contravariant tensor in defines a
-linear form on
, and if
is finite dimensional, the converse is also true.
5) Every tensor
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where and
, defines a linear transformation
of the space
given by the formula
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if , any linear transformation of the space
is defined by a tensor of type
.
6) Similarly, any tensor of type defines in
a bilinear operation, that is, the structure of a
-algebra. Moreover, if
, then any
-algebra structure in
is defined by a tensor of type
, called the structure tensor of the algebra.
Let be finite dimensional, let
be a basis of it, and let
be the dual basis of the space
. Then the tensors
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form a basis of the space . The components
of a tensor
with respect to this basis are also called the components of the tensor
with respect to the basis
of the space
. For instance, the components of a vector and of a covector coincide with their usual coordinates with respect to the bases
and
; the components of a tensor of type
coincide with the entries of the matrix corresponding to the bilinear form; the components of a tensor of type
coincide with the entries of the matrix of the corresponding linear transformation, and the components of the structure tensor of an algebra coincide with its structure constants. If
is another basis of
, with
, and
, then the components
of the tensor
in this basis are defined by the formula
![]() | (1) |
Here, as often happens in tensor calculus, Einstein's summation convention is applied: with respect to any pair of equal indices of which one is an upper index and the other is a lower index, it is understood that summation from 1 to is carried out. Conversely, if a system of
elements of a field
depending on the basis of the space
is altered in the transition from one basis to another basis according to the formulas (1), then this system is the set of components of some tensor of type
.
In the vector space the operations of addition of tensors and of multiplication of a tensor by a scalar from
are defined. Under these operations the corresponding components are added, or multiplied by the scalar. The operation of multiplying tensors of different types is also defined; it is introduced as follows. There is a natural isomorphism of vector spaces
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mapping
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to
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Consequently, for any and
the element
can be regarded as a tensor of type
and is called the tensor product of
and
. The components of the product are computed according to the formula
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Let ,
, and let the numbers
and
be fixed with
and
. Then there is a well-defined mapping
such that
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It is called contraction in the -th contravariant and the
-th covariant indices. In components, the contraction is written in the form
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For instance, the contraction of a tensor of type
is the trace of the corresponding linear transformation.
A tensor is similarly defined on an arbitrary unitary module over an associative commutative ring with a unit. The stated examples and properties of tensors are transferred, with corresponding changes, to this case, it being sometimes necessary to assume that
is a free or a finitely-generated free module.
Let a non-degenerate bilinear form be fixed in a finite-dimensional vector space
over a field
(for example,
is a Euclidean or pseudo-Euclidean space over
); in this case the form
is called a metric tensor. A metric tensor defines an isomorphism
by the formula
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Let , and let the index
,
, be fixed. Then the formula
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defines an isomorphism , called lowering of the
-th contravariant index. In other terms,
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In components, lowering an index has the form
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Similarly one defines the isomorphism of raising the -th covariant index
:
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which maps onto
. In components, raising an index is written in the form
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where . In particular, raising at first the first, and then also the remaining covariant index of the metric tensor
leads to a tensor of type
with components
(a contravariant metric tensor). Sometimes the lowered (raised) index is not moved to the first (last) place, but is written in the same place in the lower (upper) group of indices, a point being put in the empty place which arises. For instance, for
the components of the tensor
are written in the form
.
Any linear mapping of vector spaces over
defines in a natural way linear mappings
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and
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If is an isomorphism, the linear mapping
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is also defined and . The correspondence
has functorial properties. In particular, it defines a linear representation
of the group
in the space
(the tensor representation).
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) |
[2] | I.M. Gel'fand, "Lectures on linear algebra" , Interscience (1961) (Translated from Russian) |
[3] | A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian) |
[4] | M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian) |
[5] | P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) |
Tensor on a vector space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_on_a_vector_space&oldid=37606