# basic tensor

The present entry employs the terminology and notation defined and
described in the entry on tensor arrays. To keep things reasonably
self-contained we mention that the symbol ${\mathrm{T}}^{p,q}$ refers to the
vector space^{} of type $(p,q)$ tensor arrays, i.e. maps

$${I}^{p}\times {I}^{q}\to \mathbb{K},$$ |

where $I$ is some finite list of index labels, and where $\mathbb{K}$ is a field.

We say that a tensor array is a characteristic array, a.k.a. a basic tensor, if all but one of its values are $0$, and the remaining non-zero value is equal to $1$. For tuples $A\in {I}^{p}$ and $B\in {I}^{q}$, we let

$${\epsilon}_{A}^{B}:{I}^{p}\times {I}^{q}\to \mathbb{K},$$ |

denote the characteristic array defined by

$${({\epsilon}_{A}^{B})}_{{j}_{1}\mathrm{\dots}{j}_{q}}^{{i}_{1}\mathrm{\dots}{i}_{p}}=\{\begin{array}{cc}\hfill 1& \text{if}({i}_{1},\mathrm{\dots},{i}_{p})=A\text{and}({j}_{1},\mathrm{\dots},{j}_{p})=B,\hfill \\ \hfill 0& \text{otherwise.}\hfill \end{array}$$ |

The type $(p,q)$ characteristic arrays form a natural basis for ${\mathrm{T}}^{p,q}$.

Furthermore the outer multiplication of two characteristic arrays gives a characteristic array of larger valence. In other words, for

$${A}_{1}\in {I}^{{p}_{1}},{B}_{1}\in {I}^{{q}_{1}},{A}_{2}\in {I}^{{p}_{2}},{B}_{2}\in {I}^{{q}_{2}},$$ |

we have that

$${\epsilon}_{{A}_{1}}^{{B}_{1}}{\epsilon}_{{A}_{2}}^{{B}_{2}}={\epsilon}_{{A}_{1}{A}_{2}}^{{B}_{1}{B}_{2}},$$ |

where
the product^{} on the left-hand side is performed by outer
multiplication, and where ${A}_{1}{A}_{2}$ on the right-hand side refers to
the element of ${I}^{{p}_{1}+{p}_{2}}$ obtained by concatenating the tuples
${A}_{1}$ and ${A}_{2}$, and similarly for ${B}_{1}{B}_{2}$.

In this way we see that the type $(1,0)$ characteristic arrays
${\epsilon}_{(i)},i\in I$ (the natural basis of ${\mathbb{K}}^{I}$), and the type
$(0,1)$ characteristic arrays ${\epsilon}^{(i)},i\in I$ (the natural basis of
${\left({\mathbb{K}}^{I}\right)}^{*}$) generate the tensor array algebra relative to the
outer multiplication operation^{}.

The just-mentioned fact gives us an alternate way of writing and thinking about tensor arrays. We introduce the basic symbols

$${\epsilon}_{(i)},{\epsilon}^{(i)},i\in I$$ |

subject to the commutation relations^{}

$${\epsilon}_{(i)}{\epsilon}^{({i}^{\prime})}={\epsilon}^{({i}^{\prime})}{\epsilon}_{(i)},i,{i}^{\prime}\in I,$$ |

add and multiply these symbols using coefficients in $\mathbb{K}$, and use

$${\epsilon}_{({j}_{1}\mathrm{\dots}{j}_{p})}^{({i}_{1}\mathrm{\dots}{i}_{q})},{i}_{1},\mathrm{\dots},{i}_{q},{j}_{1},\mathrm{\dots},{j}_{p}\in I$$ |

as a handy abbreviation for

$${\epsilon}^{({i}_{1})}\mathrm{\dots}{\epsilon}^{({i}_{q})}{\epsilon}_{({j}_{1})}\mathrm{\dots}{\epsilon}_{({j}_{p})}.$$ |

We then interpret the resulting expressions as tensor arrays in the obvious fashion: the values of the tensor array are just the coefficients of the $\epsilon $ symbol matching the given index. However, note that in the $\epsilon $ symbols, the covariant data is written as a superscript, and the contravariant data as a subscript. This is done to facilitate the Einstein summation convention.

By way of illustration, suppose that $I=(1,2)$. We can now write down
a type $(1,0)$ tensor, i.e. a column vector^{}

$$u=\left(\begin{array}{c}\hfill {u}^{1}\hfill \\ \hfill {u}^{2}\hfill \end{array}\right)\in {\mathrm{T}}^{1,0}$$ |

as

$$u={u}^{1}{\epsilon}_{(1)}+{u}^{2}{\epsilon}_{(2)}.$$ |

Similarly, a row-vector

$$\varphi =({\varphi}_{1},{\varphi}_{2})\in {\mathrm{T}}^{0,1}$$ |

can be written down as

$$\varphi ={\varphi}_{1}{\epsilon}^{(1)}+{\varphi}_{2}{\epsilon}^{(2)}.$$ |

In the case of a matrix

$$M=\left(\begin{array}{cc}\hfill {M}_{1}^{1}\hfill & \hfill {M}_{1}^{2}\hfill \\ \hfill {M}_{2}^{1}\hfill & \hfill {M}_{2}^{2}\hfill \end{array}\right)\in {\mathrm{T}}^{1,1}$$ |

we would write

$$M={M}_{1}^{1}{\epsilon}_{(1)}^{(1)}+{M}_{2}^{1}{\epsilon}_{(1)}^{(2)}+{M}_{1}^{2}{\epsilon}_{(2)}^{(1)}+{M}_{2}^{2}{\epsilon}_{(2)}^{(2)}.$$ |

Title | basic tensor |
---|---|

Canonical name | BasicTensor |

Date of creation | 2013-03-22 12:40:37 |

Last modified on | 2013-03-22 12:40:37 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 7 |

Author | rmilson (146) |

Entry type | Derivation |

Classification | msc 15A69 |

Synonym | characteristic array |

Related topic | TensorArray |

Related topic | Basis |

Related topic | Frame |

Related topic | SimpleTensor |