# 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

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

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

 $\varepsilon^{B}_{A}:I^{p}\times I^{q}\rightarrow\mathbb{K},$

denote the characteristic array defined by

 $(\varepsilon^{B}_{A})^{i_{1}\ldots i_{p}}_{j_{1}\ldots j_{q}}=\left\{\begin{% array}[]{rl}1&\mbox{ if (i_{1},\ldots,i_{p})=A and (j_{1},\ldots,j_{p})=B}% ,\\ 0&\mbox{ otherwise.}\end{array}\right.$

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

 $\varepsilon^{B_{1}}_{A_{1}}\varepsilon^{B_{2}}_{A_{2}}=\varepsilon^{B_{1}B_{2}% }_{A_{1}A_{2}},$

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 $\varepsilon_{(i)},\;i\in I$ (the natural basis of $\mathbb{K}^{I}$), and the type $(0,1)$ characteristic arrays $\varepsilon^{(i)},\;i\in I$ (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

 $\varepsilon_{(i)},\;\varepsilon^{(i)},\quad i\in I$
 $\varepsilon_{(i)}\varepsilon^{(i^{\prime})}=\varepsilon^{(i^{\prime})}% \varepsilon_{(i)},\quad i,i^{\prime}\in I,$

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

 $\varepsilon^{(i_{1}\ldots i_{q})}_{(j_{1}\ldots j_{p})},\quad i_{1},\ldots,i_{% q},j_{1},\ldots,j_{p}\in I$

as a handy abbreviation for

 $\varepsilon^{(i_{1})}\ldots\varepsilon^{(i_{q})}\varepsilon_{(j_{1})}\ldots% \varepsilon_{(j_{p})}.$

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 $\varepsilon$ symbol matching the given index. However, note that in the $\varepsilon$ 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  $u=\begin{pmatrix}u^{1}\\ u^{2}\end{pmatrix}\in\mathrm{T}^{1,0}$

as

 $u=u^{1}\varepsilon_{(1)}+u^{2}\varepsilon_{(2)}.$

Similarly, a row-vector

 $\phi=(\phi_{1},\phi_{2})\in\mathrm{T}^{0,1}$

can be written down as

 $\phi=\phi_{1}\varepsilon^{(1)}+\phi_{2}\varepsilon^{(2)}.$

In the case of a matrix

 $M=\begin{pmatrix}M^{1}_{\!\hphantom{1}1}&M^{2}_{\!\hphantom{2}1}\\ M^{1}_{\!\hphantom{1}2}&M^{2}_{\!\hphantom{2}2}\end{pmatrix}\in\mathrm{T}^{1,1}$

we would write

 $M=M^{1}_{\!\hphantom{1}1}\,\varepsilon^{(1)}_{(1)}+M^{1}_{\!\hphantom{1}2}\,% \varepsilon^{(2)}_{(1)}+M^{2}_{\!\hphantom{2}1}\,\varepsilon^{(1)}_{(2)}+M^{2}% _{\!\hphantom{2}2}\,\varepsilon^{(2)}_{(2)}.$
Title basic tensor BasicTensor 2013-03-22 12:40:37 2013-03-22 12:40:37 rmilson (146) rmilson (146) 7 rmilson (146) Derivation msc 15A69 characteristic array TensorArray Basis Frame SimpleTensor