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[parent] outer multiplication (Definition)

Note: 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.

Let $ p_1,p_2,q_1,q_2$ be natural numbers. Outer multiplication is a bilinear operation

$\displaystyle \mathrm{T}^{p_1,q_1} \times \mathrm{T}^{p_2,q_2} \rightarrow \mathrm{T}^{p_1+p_2,q_1+q_2}$
that combines a type $ (p_1,q_1)$ tensor array $ X$ and a type $ (p_2,q_2)$ tensor array $ Y$ to produce a type $ (p_1+p_2,q_1+q_2)$ tensor array $ XY$ (also written as $ X\otimes Y$), defined by
$\displaystyle (XY)^{i_1\ldots i_{p_1} i_{p_1+1} \ldots i_{p_1+p_2}}_{j_1\ldots ... ...1\ldots j_{q_1}} Y^{i_{p_1+1}\ldots i_{p_1+p_2}}_{j_{q_1+1}\ldots j_{q_1+q_2}} $
Speaking informally, what is going on above is that we multiply every value of the $ X$ array by every possible value of the $ Y$ array, to create a new array, $ XY$. Quite obviously then, the size of $ XY$ is the size of $ X$ times the size of $ Y$, and the index slots of the product $ XY$ are just the union of the index slots of $ X$ and of $ Y$.

Outer multiplication is a non-commutative, associative operation. The type $ (0,0)$ arrays are the scalars, i.e. elements of $ \mathbb{K}$; they commute with everything. Thus, we can embed $ \mathbb{K}$ into the direct sum

$\displaystyle \bigoplus_{p,q\in\mathbb{N}} \mathrm{T}^{p,q},$
and thereby endow the latter with the structure of an $ \mathbb{K}$-algebra 1.

By way of illustration we mention that the outer product of a column vector, i.e. a type $ (1,0)$ array, and a row vector, i.e. a type $ (0,1)$ array, gives a matrix, i.e. a type $ (1,1)$ tensor array. For instance:

$\displaystyle \begin{pmatrix} a \\ b \\ c \end{pmatrix}\otimes \begin{pmatrix} ... ... bx & by & bz \ cx & cy & cz \end{pmatrix},\quad a,b,c,x,y,z\in \mathbb{K} $



Footnotes

...-algebra 1
We will not pursue this line of thought here, because the topic of algebra structure is best dealt with in the a more abstract context. The same comment applies to the use of the tensor product sign $ \otimes$ in denoting outer multiplication. These topics are dealt with in the entry pertaining to abstract tensor algebra.


"outer multiplication" is owned by rmilson.
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See Also: tensor product (vector spaces), tensor product


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Cross-references: matrix, row vector, column vector, outer product, tensor algebra, tensor product, algebra, line, structure, direct sum, scalars, operation, associative, non-commutative, union, product, size, bilinear operation, natural numbers, field, labels, index, finite, maps, type, vector space, contained, tensor arrays
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This is version 1 of outer multiplication, born on 2002-05-27.
Object id is 2951, canonical name is OuterMultiplication.
Accessed 5878 times total.

Classification:
AMS MSC15A69 (Linear and multilinear algebra; matrix theory :: Multilinear algebra, tensor products)

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