Tensor on a vector space

$ V $ over a field $ k $

An element $ t $ of the vector space

$$ T ^ {p,q} ( V) = \ \left ( \otimes ^ { p } V \right ) \otimes \ \left ( \otimes ^ { q } V ^ {*} \right ) , $$

where $ V ^ {*} = \mathop{\rm Hom} ( V, k) $ is the dual space of $ V $. The tensor $ t $ is said to be $ p $ times contravariant and $ q $ times covariant, or to be of type $ ( p, q) $. The number $ p $ is called the contravariant valency, and $ q $ the covariant valency, while the number $ p + q $ is called the general valency of the tensor $ t $. The space $ T ^ {0,0} ( V) $ is identified with $ k $. Tensors of type $ ( p, 0) $ are called contravariant, those of the type $ ( 0, q) $ are called covariant, and the remaining ones are called mixed.

Examples of tensors.

1) A vector of the space $ V $( a tensor of type $ ( 1, 0) $).

2) A covector of the space $ V $( a tensor of type $ ( 0, 1) $).

3) Any covariant tensor

$$ t = \ \sum _ {i = 1 } ^ { s } h _ {i1} \otimes \dots \otimes h _ {iq} , $$

where $ h _ {ij} \in V ^ {*} $, defines a $ q $- linear form $ \widehat{t} $ on $ V $ by the formula

$$ \widehat{t} ( x _ {1} \dots x _ {q} ) = \ \sum _ {i = 1 } ^ { s } h _ {i1} ( x _ {1} ) \dots h _ {iq} ( x _ {q} ); $$

the mapping $ t \mapsto \widehat{t} $ from the space $ T ^ {0,q} $ into the space $ L ^ {q} ( V) $ of all $ q $- linear forms on $ V $ is linear and injective; if $ \mathop{\rm dim} V < \infty $, then this mapping is an isomorphism, since any $ q $- linear form corresponds to some tensor of type $ ( 0, q) $.

4) Similarly, a contravariant tensor in $ T ^ {p,0} ( V) $ defines a $ p $- linear form on $ V ^ {*} $, and if $ V $ is finite dimensional, the converse is also true.

5) Every tensor

$$ t = \ \sum _ {i = 1 } ^ { s } x _ {i} \otimes h _ {i} \ \in T ^ {1,1} ( V), $$

where $ x _ {i} \in V $ and $ h _ {j} \in V ^ {*} $, defines a linear transformation $ \widehat{t} $ of the space $ V $ given by the formula

$$ \widehat{t} ( y) = \ \sum _ {i = 1 } ^ { s } h _ {i} ( y) x _ {i} ; $$

if $ \mathop{\rm dim} V < \infty $, any linear transformation of the space $ V $ is defined by a tensor of type $ ( 1, 1) $.

6) Similarly, any tensor of type $ ( 1, 2) $ defines in $ V $ a bilinear operation, that is, the structure of a $ k $- algebra. Moreover, if $ \mathop{\rm dim} V < \infty $, then any $ k $- algebra structure in $ V $ is defined by a tensor of type $ ( 1, 2) $, called the structure tensor of the algebra.

Let $ V $ be finite dimensional, let $ v _ {1} \dots v _ {n} $ be a basis of it, and let $ v ^ {1} \dots v ^ {n} $ be the dual basis of the space $ V ^ {*} $. Then the tensors

$$ v _ {i _ {1} \dots i _ {p} } ^ {i _ {1} \dots i _ {q} } = \ v _ {i _ {1} } \otimes \dots \otimes v _ {i _ {p} } \otimes v ^ {j _ {1} } \otimes \dots \otimes v ^ {j _ {q} } $$

form a basis of the space $ T ^ {p,q} ( V) $. The components $ t _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $ of a tensor $ t \in T ^ {p,q} ( V) $ with respect to this basis are also called the components of the tensor $ t $ with respect to the basis $ v _ {1} \dots v _ {n} $ of the space $ V $. For instance, the components of a vector and of a covector coincide with their usual coordinates with respect to the bases $ ( v _ {i} ) $ and $ ( v ^ {j} ) $; the components of a tensor of type $ ( 0, 2) $ coincide with the entries of the matrix corresponding to the bilinear form; the components of a tensor of type $ ( 1, 1) $ 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 $ \widetilde{v} _ {1} \dots \widetilde{v} _ {n} $ is another basis of $ V $, with $ \widetilde{v} _ {j} = a _ {j} ^ {i} v _ {i} $, and $ \| b _ {j} ^ {i} \| = ( \| a _ {j} ^ {i} \| ^ {T} ) ^ {-} 1 $, then the components $ t _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $ of the tensor $ t $ in this basis are defined by the formula

$$ \tag{1 } \widetilde{t} {} _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots j _ {p} } = \ b _ {k _ {1} } ^ {i _ {1} } \dots b _ {k _ {p} } ^ {i _ {p} } a _ {j _ {1} } ^ {l _ {1} } \dots a _ {j _ {q} } ^ {l _ {q} } t _ {l _ {1} \dots l _ {q} } ^ {k _ {1} \dots k _ {p} } . $$

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 $ n $ is carried out. Conversely, if a system of $ n ^ {p + q } $ elements of a field $ k $ depending on the basis of the space $ V $ 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 $ ( p, q) $.

In the vector space $ T ^ {p,q} ( V) $ the operations of addition of tensors and of multiplication of a tensor by a scalar from $ k $ 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

$$ T ^ {p,q} ( V) \otimes T ^ {r,s} ( V) \cong \ T ^ {p + r, q + s } ( V), $$

mapping

$$ ( x _ {1} \otimes \dots \otimes x _ {p} \otimes h _ {1} \otimes {} \dots \otimes h _ {q} ) \otimes $$

$$ \otimes ( x _ {1} ^ \prime \otimes \dots \otimes x _ {r} ^ \prime \otimes h _ {1} ^ \prime \otimes \dots \otimes h _ {s} ) $$

to

$$ x _ {1} \otimes \dots \otimes x _ {p} \otimes x _ {1} ^ \prime \otimes \dots \otimes x _ {r} ^ \prime \otimes $$

$$ \otimes h _ {1} \otimes \dots \otimes h _ {q} \otimes h _ {1} ^ \prime \otimes \dots \otimes h _ {s} ^ \prime . $$

Consequently, for any $ t \in T ^ {p,q} ( V) $ and $ u \in T ^ {r,s} ( V) $ the element $ v = t \otimes u $ can be regarded as a tensor of type $ ( p + r, q + s) $ and is called the tensor product of $ t $ and $ u $. The components of the product are computed according to the formula

$$ v _ {j _ {1} \dots j _ {q + s } } ^ {i _ {1} \dots i _ {p + r } } = \ t _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } u _ {j _ {q + 1 } \dots j _ {q + s } } ^ {i _ {p + 1 } \dots i _ {p + r } } . $$

Let $ p > 0 $, $ q > 0 $, and let the numbers $ \alpha $ and $ \beta $ be fixed with $ 1 \leq \alpha \leq p $ and $ 1 \leq \beta \leq q $. Then there is a well-defined mapping $ Y _ \beta ^ \alpha : T ^ {p,q} ( V) \rightarrow T ^ {p - 1, q - 1 } ( V) $ such that

$$ Y _ \beta ^ \alpha ( x _ {1} \otimes \dots \otimes x _ {p} \otimes h _ {1} \otimes \dots \otimes h _ {q} ) = $$

$$ = \ h _ \beta ( x _ \alpha ) x _ {1} \otimes \dots \otimes x _ {\alpha - 1 } \otimes x _ {\alpha + 1 } \otimes \dots \otimes x _ {p} \otimes $$

$$ \otimes h _ {1} \otimes \dots \otimes h _ {\beta - 1 } \otimes h _ {\beta + 1 } \otimes \dots \otimes h _ {q} . $$

It is called contraction in the $ \alpha $- th contravariant and the $ \beta $- th covariant indices. In components, the contraction is written in the form

$$ ( Y _ \beta ^ \alpha t) _ {j _ {1} \dots j _ {q - 1 } } ^ {i _ {1} \dots i _ {p - 1 } } = \ t _ {j _ {1} \dots j _ {\beta - 1 } ij _ {\beta + 1 } \dots j _ {q} } ^ {i _ {1} \dots i _ {\alpha - 1 } ii _ {\alpha + 1 } \dots i _ {p} } . $$

For instance, the contraction $ Y _ {1} ^ {1} t $ of a tensor of type $ ( 1, 1) $ is the trace of the corresponding linear transformation.

A tensor is similarly defined on an arbitrary unitary module $ V $ 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 $ V $ is a free or a finitely-generated free module.

Let a non-degenerate bilinear form $ g $ be fixed in a finite-dimensional vector space $ V $ over a field $ k $( for example, $ V $ is a Euclidean or pseudo-Euclidean space over $ \mathbf R $); in this case the form $ g $ is called a metric tensor. A metric tensor defines an isomorphism $ \gamma : V \rightarrow V ^ {*} $ by the formula

$$ \gamma ( x) ( y) = g ( x, y),\ \ x, y \in V. $$

Let $ p > 0 $, and let the index $ \alpha $, $ 1 \leq \alpha \leq p $, be fixed. Then the formula

$$ x _ {1} \otimes \dots \otimes x _ {p} \otimes h _ {1} \otimes \dots \otimes h _ {q\ } \mapsto $$

$$ \mapsto \ x _ {1} \otimes \dots \otimes x _ {\alpha - 1 } \otimes x _ {\alpha + 1 } \otimes \dots \otimes x _ {p} \otimes $$

$$ \otimes \gamma ( x _ \alpha ) \otimes h _ {1} \otimes \dots \otimes h _ {q} $$

defines an isomorphism $ \gamma ^ \alpha : T ^ {p,q} ( V) \rightarrow T ^ {p - 1, q + 1 } ( V) $, called lowering of the $ \alpha $- th contravariant index. In other terms,

$$ \gamma ^ \alpha ( t) = \ Y _ {1} ^ \alpha ( g \otimes t). $$

In components, lowering an index has the form

$$ \gamma ^ \alpha ( t) _ {j _ {1} \dots j _ {q + 1 } } ^ {i _ {1} \dots i _ {q - 1 } } = \ g _ { ij _ 1 } t _ {j _ {2} \dots j _ {q + 1 } } ^ {i _ {1} \dots i _ {\alpha - 1 } ii _ {\alpha + 1 } \dots i _ {p - 1 } } . $$

Similarly one defines the isomorphism of raising the $ \beta $- th covariant index $ ( 1 \leq \beta \leq q) $:

$$ \gamma _ \beta : \ x _ {1} \otimes \dots \otimes x _ {p} \otimes h _ {1} \otimes \dots \otimes h _ {q\ } \mapsto $$

$$ \mapsto \ x _ {1} \otimes \dots \otimes x _ {p} \otimes \gamma ^ {-} 1 ( h _ \beta ) \otimes $$

$$ \otimes h _ {1} \otimes \dots \otimes h _ {\beta - 1 } \otimes h _ {\beta + 1 } \otimes \dots \otimes h _ {q} , $$

which maps $ T ^ {p,q} ( V) $ onto $ T ^ {p + 1, q - 1 } ( V) $. In components, raising an index is written in the form

$$ \gamma _ \beta ( t) _ {j _ {1} \dots j _ {q - 1 } } ^ {i _ {1} \dots i _ {p + 1 } } = \ g ^ {ji _ {p + 1 } } t _ {j _ {1} \dots j _ {\beta - 1 } ij _ \beta \dots j _ {q - 1 } } ^ {i _ {1} \dots i _ {p} } , $$

where $ \| g ^ {kl} \| = (\| g _ {ij} \| ^ {T} ) ^ {-} 1 $. In particular, raising at first the first, and then also the remaining covariant index of the metric tensor $ g $ leads to a tensor of type $ ( 2, 0) $ with components $ g ^ {kl} $( 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 $ t \in T ^ {2,0} ( V) $ the components of the tensor $ \gamma ^ {2} ( t) $ are written in the form $ t _ {j} ^ {i. } = g _ {kj} t ^ {ik} $.

Any linear mapping $ f: V \rightarrow W $ of vector spaces over $ k $ defines in a natural way linear mappings

$$ T ^ {p,0} ( f ) = \ \otimes ^ { p } f: \ T ^ {p,0} ( V) \rightarrow T ^ {p,0} ( W) $$

and

$$ T ^ {q,0} ( f ^ { * } ) = \ \otimes ^ { q } f ^ { * } : \ T ^ {0,q} ( W) \rightarrow T ^ {0,q} ( V). $$

If $ f $ is an isomorphism, the linear mapping

$$ T ^ {p,q} ( f ): \ T ^ {p,q} ( V) \rightarrow T ^ {p,q} ( W) $$

is also defined and $ T ^ {0,q} ( f ) = T ^ {q,0} ( f ^ { * } ) ^ {-} 1 $. The correspondence $ f \mapsto T ^ {p,q} ( f ) $ has functorial properties. In particular, it defines a linear representation $ a \mapsto T ^ {p,q} ( a) $ of the group $ \mathop{\rm GL} ( V) $ in the space $ T ^ {p,q} ( V) $( 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)