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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924001.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924002.png" />''
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An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924003.png" /> of the vector space
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924004.png" /></td> </tr></table>
+
'' $  V $
 +
over a field  $  k $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924005.png" /> is the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924006.png" />. The tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924007.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924008.png" /> times contravariant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t0924009.png" /> times covariant, or to be of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240011.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240012.png" /> is called the contravariant valency, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240013.png" /> the covariant valency, while the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240014.png" /> is called the general valency of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240015.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240016.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240017.png" />. Tensors of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240018.png" /> are called contravariant, those of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240019.png" /> are called covariant, and the remaining ones are called mixed.
+
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.
 
Examples of tensors.
  
1) A vector of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240020.png" /> (a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240021.png" />).
+
1) A vector of the space $  V $(
 +
a tensor of type $  ( 1, 0) $).
  
2) A covector of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240022.png" /> (a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240023.png" />).
+
2) A covector of the space $  V $(
 +
a tensor of type $  ( 0, 1) $).
  
 
3) Any covariant tensor
 
3) Any covariant tensor
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240024.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {i = 1 } ^ { s }
 +
h _ {i1} \otimes \dots \otimes h _ {iq} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240025.png" />, defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240026.png" />-linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240028.png" /> by the formula
+
where $  h _ {ij} \in V  ^ {*} $,  
 +
defines a $  q $-
 +
linear form $  \widehat{t}  $
 +
on $  V $
 +
by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240029.png" /></td> </tr></table>
+
$$
 +
\widehat{t}  ( x _ {1} \dots x _ {q} )  = \
 +
\sum _ {i = 1 } ^ { s }
 +
h _ {i1} ( x _ {1} ) \dots
 +
h _ {iq} ( x _ {q} );
 +
$$
  
the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240030.png" /> from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240031.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240032.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240033.png" />-linear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240034.png" /> is linear and injective; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240035.png" />, then this mapping is an isomorphism, since any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240036.png" />-linear form corresponds to some tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240037.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240038.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240039.png" />-linear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240040.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240041.png" /> is finite dimensional, the converse is also true.
+
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
 
5) Every tensor
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240042.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {i = 1 } ^ { s }
 +
x _ {i} \otimes h _ {i} \
 +
\in  T  ^ {1,1} ( V),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240044.png" />, defines a linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240045.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240046.png" /> given by the formula
+
where $  x _ {i} \in V $
 +
and $  h _ {j} \in V  ^ {*} $,  
 +
defines a linear transformation $  \widehat{t}  $
 +
of the space $  V $
 +
given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240047.png" /></td> </tr></table>
+
$$
 +
\widehat{t}  ( y)  = \
 +
\sum _ {i = 1 } ^ { s }
 +
h _ {i} ( y) x _ {i} ;
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240048.png" />, any linear transformation of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240049.png" /> is defined by a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240050.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240051.png" /> defines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240052.png" /> a bilinear operation, that is, the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240053.png" />-algebra. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240054.png" />, then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240056.png" />-algebra structure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240057.png" /> is defined by a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240058.png" />, called the structure tensor of the algebra.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240059.png" /> be finite dimensional, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240060.png" /> be a basis of it, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240061.png" /> be the dual basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240062.png" />. Then the tensors
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240063.png" /></td> </tr></table>
+
$$
 +
v _ {i _ {1}  \dots i _ {p} } ^ {i _ {1} \dots i _ {q} }  = \
 +
v _ {i _ {1}  } \otimes \dots \otimes v _ {i _ {p}  } \otimes
 +
v ^ {j _ {1} } \otimes \dots \otimes v ^ {j _ {q} }
 +
$$
  
form a basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240064.png" />. The components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240065.png" /> of a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240066.png" /> with respect to this basis are also called the components of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240067.png" /> with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240068.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240069.png" />. For instance, the components of a vector and of a covector coincide with their usual coordinates with respect to the bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240071.png" />; the components of a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240072.png" /> coincide with the entries of the matrix corresponding to the bilinear form; the components of a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240073.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240074.png" /> is another basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240075.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240076.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240077.png" />, then the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240078.png" /> of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240079.png" /> in this basis are defined by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\widetilde{t}  {} _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots j _ {p} }  = \
 +
b _ {k _ {1}  } ^ {i _ {1} } \dots
 +
b _ {k _ {p}  } ^ {i _ {p} }
 +
a _ {j _ {1}  } ^ {l _ {1} } \dots
 +
a _ {j _ {q}  } ^ {l _ {q} }
 +
t _ {l _ {1}  \dots l _ {q} } ^ {k _ {1} \dots k _ {p} } .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240081.png" /> is carried out. Conversely, if a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240082.png" /> elements of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240083.png" /> depending on the basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240084.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240085.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240086.png" /> the operations of addition of tensors and of multiplication of a tensor by a scalar from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240087.png" /> 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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240088.png" /></td> </tr></table>
+
$$
 +
T  ^ {p,q} ( V) \otimes
 +
T  ^ {r,s} ( V)  \cong \
 +
T ^ {p + r, q + s } ( V),
 +
$$
  
 
mapping
 
mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240089.png" /></td> </tr></table>
+
$$
 +
( x _ {1} \otimes \dots \otimes
 +
x _ {p} \otimes h _ {1} \otimes
 +
{} \dots \otimes h _ {q} ) \otimes
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240090.png" /></td> </tr></table>
+
$$
 +
\otimes
 +
( x _ {1}  ^  \prime  \otimes \dots
 +
\otimes x _ {r}  ^  \prime  \otimes
 +
h _ {1}  ^  \prime  \otimes \dots \otimes h _ {s} )
 +
$$
  
 
to
 
to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240091.png" /></td> </tr></table>
+
$$
 +
x _ {1} \otimes \dots \otimes
 +
x _ {p} \otimes
 +
x _ {1}  ^  \prime  \otimes \dots
 +
\otimes x _ {r}  ^  \prime  \otimes
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240092.png" /></td> </tr></table>
+
$$
 +
\otimes
 +
h _ {1} \otimes \dots
 +
\otimes h _ {q} \otimes
 +
h _ {1}  ^  \prime  \otimes \dots
 +
\otimes h _ {s}  ^  \prime  .
 +
$$
  
Consequently, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240094.png" /> the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240095.png" /> can be regarded as a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240096.png" /> and is called the tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240098.png" />. The components of the product are computed according to the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t09240099.png" /></td> </tr></table>
+
$$
 +
v _ {j _ {1}  \dots j _ {q + s }  } ^ {i _ {1} \dots i _ {p + r }  }  = \
 +
t _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }
 +
u _ {j _ {q + 1 }  \dots j _ {q + s }  } ^ {i _ {p + 1 }  \dots i _ {p + r }  } .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400101.png" />, and let the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400103.png" /> be fixed with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400105.png" />. Then there is a well-defined mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400106.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400107.png" /></td> </tr></table>
+
$$
 +
Y _  \beta  ^  \alpha
 +
( x _ {1} \otimes \dots \otimes x _ {p} \otimes
 +
h _ {1} \otimes \dots \otimes h _ {q} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400108.png" /></td> </tr></table>
+
$$
 +
= \
 +
h _  \beta  ( x _  \alpha  ) x _ {1} \otimes \dots
 +
\otimes x _ {\alpha - 1 }
 +
\otimes x _ {\alpha + 1 }
 +
\otimes \dots \otimes
 +
x _ {p} \otimes
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400109.png" /></td> </tr></table>
+
$$
 +
\otimes
 +
h _ {1} \otimes \dots \otimes
 +
h _ {\beta - 1 }
 +
\otimes h _ {\beta + 1 }
 +
\otimes \dots \otimes h _ {q} .
 +
$$
  
It is called contraction in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400110.png" />-th contravariant and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400111.png" />-th covariant indices. In components, the contraction is written in the form
+
It is called contraction in the $  \alpha $-
 +
th contravariant and the $  \beta $-
 +
th covariant indices. In components, the contraction is written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400112.png" /></td> </tr></table>
+
$$
 +
( Y _  \beta  ^  \alpha  t) _ {j _ {1}  \dots j _ {q - 1 }  } ^ {i _ {1} \dots i _ {p - 1 }  }  = \
 +
t _ {j _ {1}  \dots j _ {\beta - 1 }
 +
ij _ {\beta + 1 }  \dots j _ {q} } ^ {i _ {1} \dots i _ {\alpha - 1 }
 +
ii _ {\alpha + 1 }  \dots i _ {p} } .
 +
$$
  
For instance, the contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400113.png" /> of a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400114.png" /> is the trace of the corresponding linear transformation.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400115.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400116.png" /> is a free or a finitely-generated free module.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400117.png" /> be fixed in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400118.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400119.png" /> (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400120.png" /> is a Euclidean or pseudo-Euclidean space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400121.png" />); in this case the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400122.png" /> is called a metric tensor. A metric tensor defines an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400123.png" /> by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400124.png" /></td> </tr></table>
+
$$
 +
\gamma ( x) ( y)  = g ( x, y),\ \
 +
x, y \in V.
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400125.png" />, and let the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400127.png" />, be fixed. Then the formula
+
Let $  p > 0 $,  
 +
and let the index $  \alpha $,  
 +
$  1 \leq  \alpha \leq  p $,  
 +
be fixed. Then the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400128.png" /></td> </tr></table>
+
$$
 +
x _ {1} \otimes \dots \otimes
 +
x _ {p} \otimes
 +
h _ {1} \otimes \dots \otimes
 +
h _ {q\ } \mapsto
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400129.png" /></td> </tr></table>
+
$$
 +
\mapsto \
 +
x _ {1} \otimes \dots \otimes
 +
x _ {\alpha - 1 }  \otimes x _ {\alpha + 1 }  \otimes \dots
 +
\otimes x _ {p} \otimes
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400130.png" /></td> </tr></table>
+
$$
 +
\otimes
 +
\gamma ( x _  \alpha  )
 +
\otimes h _ {1} \otimes \dots
 +
\otimes h _ {q}  $$
  
defines an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400131.png" />, called lowering of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400132.png" />-th contravariant index. In other terms,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400133.png" /></td> </tr></table>
+
$$
 +
\gamma  ^  \alpha  ( t)  = \
 +
Y _ {1}  ^  \alpha
 +
( g \otimes t).
 +
$$
  
 
In components, lowering an index has the form
 
In components, lowering an index has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400134.png" /></td> </tr></table>
+
$$
 +
\gamma  ^  \alpha  ( t) _ {j _ {1}  \dots j _ {q + 1 }  } ^ {i _ {1} \dots i _ {q - 1 }  }  = \
 +
g _ { ij _ 1 } t _ {j _ {2}  \dots j _ {q + 1 }  } ^ {i _ {1} \dots i _ {\alpha - 1 }
 +
ii _ {\alpha + 1 }  \dots i _ {p - 1 }  } .
 +
$$
  
Similarly one defines the isomorphism of raising the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400135.png" />-th covariant index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400136.png" />:
+
Similarly one defines the isomorphism of raising the $  \beta $-
 +
th covariant index $  ( 1 \leq  \beta \leq  q) $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400137.png" /></td> </tr></table>
+
$$
 +
\gamma _  \beta  : \
 +
x _ {1} \otimes \dots \otimes
 +
x _ {p} \otimes
 +
h _ {1} \otimes \dots \otimes
 +
h _ {q\ } \mapsto
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400138.png" /></td> </tr></table>
+
$$
 +
\mapsto \
 +
x _ {1} \otimes \dots \otimes
 +
x _ {p} \otimes \gamma  ^ {-} 1 ( h _  \beta  ) \otimes
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400139.png" /></td> </tr></table>
+
$$
 +
\otimes
 +
h _ {1} \otimes \dots \otimes
 +
h _ {\beta - 1 }  \otimes h _ {\beta + 1 }
 +
\otimes \dots \otimes h _ {q} ,
 +
$$
  
which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400140.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400141.png" />. In components, raising an index is written in the form
+
which maps $  T  ^ {p,q} ( V) $
 +
onto $  T ^ {p + 1, q - 1 } ( V) $.  
 +
In components, raising an index is written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400142.png" /></td> </tr></table>
+
$$
 +
\gamma _  \beta  ( t) _ {j _ {1}  \dots j _ {q - 1 }  } ^ {i _ {1} \dots i _ {p + 1 }  }  = \
 +
g ^ {ji _ {p + 1 }  }
 +
t _ {j _ {1}  \dots j _ {\beta - 1 }
 +
ij _  \beta  \dots j _ {q - 1 }  } ^ {i _ {1} \dots i _ {p} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400143.png" />. In particular, raising at first the first, and then also the remaining covariant index of the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400144.png" /> leads to a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400145.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400146.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400147.png" /> the components of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400148.png" /> are written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400149.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400150.png" /> of vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400151.png" /> defines in a natural way linear mappings
+
Any linear mapping $  f: V \rightarrow W $
 +
of vector spaces over $  k $
 +
defines in a natural way linear mappings
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400152.png" /></td> </tr></table>
+
$$
 +
T  ^ {p,0} ( f  )  = \
 +
\otimes ^ { p }  f: \
 +
T  ^ {p,0} ( V)  \rightarrow  T  ^ {p,0} ( W)
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400153.png" /></td> </tr></table>
+
$$
 +
T  ^ {q,0} ( f ^ { * } )  = \
 +
\otimes ^ { q }  f ^ { * } : \
 +
T  ^ {0,q} ( W)  \rightarrow  T  ^ {0,q} ( V).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400154.png" /> is an isomorphism, the linear mapping
+
If $  f $
 +
is an isomorphism, the linear mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400155.png" /></td> </tr></table>
+
$$
 +
T  ^ {p,q} ( f  ): \
 +
T  ^ {p,q} ( V)  \rightarrow  T  ^ {p,q} ( W)
 +
$$
  
is also defined and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400156.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400157.png" /> has functorial properties. In particular, it defines a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400158.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400159.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092400/t092400160.png" /> (the tensor representation).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  "Lectures on linear algebra" , Interscience  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Postnikov,  "Linear algebra and differential geometry" , Moscow  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  "Lectures on linear algebra" , Interscience  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Postnikov,  "Linear algebra and differential geometry" , Moscow  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


$ 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)
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=37606
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article