# 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).

How to Cite This Entry:
Tensor on a vector space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_on_a_vector_space&oldid=48957
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article