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[parent] basic tensor (Derivation)

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

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

$\displaystyle \varepsilon ^B_A:I^p\times I^q\rightarrow\mathbb{K},$
denote the characteristic array defined by
\begin{displaymath}(\varepsilon ^B_A)^{i_1\ldots i_p}_{j_1\ldots j_q} = \left\{ ... ...\ldots, j_p)=B$},\ 0 & \mbox{ otherwise.} \end{array}\right. \end{displaymath}
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

$\displaystyle 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
$\displaystyle \varepsilon ^{B_1}_{A_1} \varepsilon ^{B_2}_{A_2} = \varepsilon ^{B_1 B_2}_{A_1 A_2},$
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

$\displaystyle \varepsilon _{(i)},\; \varepsilon ^{(i)} ,\quad i\in I$
subject to the commutation relations
$\displaystyle \varepsilon _{(i)} \varepsilon ^{(i')} = \varepsilon ^{(i')}\varepsilon _{(i)} ,\quad i, i'\in I,$
add and multiply these symbols using coefficients in $ \mathbb{K}$, and use
$\displaystyle \varepsilon ^{(i_1 \ldots i_q)}_{(j_1 \ldots j_p)},\quad i_1,\ldots,i_q,j_1,\ldots,j_p\in I$
as a handy abbreviation for
$\displaystyle \varepsilon ^{(i_1)} \ldots \varepsilon ^{(i_q)} \varepsilon _{(j_1)} \ldots \varepsilon _{(j_p)}.$
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

$\displaystyle u= \begin{pmatrix} u^1 \\ u^2 \end{pmatrix}\in \mathrm{T}^{1,0} $
as
$\displaystyle u = u^1 \varepsilon _{(1)} + u^2 \varepsilon _{(2)}.$
Similarly, a row-vector
$\displaystyle \phi = (\phi_1,\phi_2) \in \mathrm{T}^{0,1}$
can be written down as
$\displaystyle \phi = \phi_1 \varepsilon ^{(1)} + \phi_2 \varepsilon ^{(2)}.$
In the case of a matrix
$\displaystyle M = \begin{pmatrix} M^{1}_{\!\hphantom{1}1} & M^{2}_{\!\hphantom{... ..._{\!\hphantom{1}2} & M^{2}_{\!\hphantom{2}2} \end{pmatrix}\in \mathrm{T}^{1,1} $
we would write
$\displaystyle M = M^{1}_{\!\hphantom{1}1}\, \varepsilon ^{(1)}_{(1)}+ M^{1}_{\!... ... \varepsilon ^{(1)}_{(2)}+ M^{2}_{\!\hphantom{2}2}\, \varepsilon ^{(2)}_{(2)}. $



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See Also: tensor array, basis, frame, simple tensor

Other names:  characteristic array

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Cross-references: matrix, column vector, tensor, Einstein summation convention, subscript, superscript, matching, obvious, expressions, coefficients, relations, operation, algebra, generate, side, product, valence, outer multiplication, basis, tuples, field, labels, index, finite, maps, type, vector space, tensor arrays
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This is version 4 of basic tensor, born on 2002-05-27, modified 2005-08-04.
Object id is 2953, canonical name is BasicTensor.
Accessed 5650 times total.

Classification:
AMS MSC: 15A69 (Linear and multilinear algebra; matrix theory :: Multilinear algebra, tensor products)
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