PlanetMath: outer multiplication

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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 tensor arrays, i.e. maps

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

where

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

Let 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

tensor array

and a type

tensor array

to produce a type

tensor array

(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

array by every possible value of the

array, to create a new array,

. Quite obviously then, the size of

is the size of

times the size of

, and the index slots of the product

are just the union of the index slots of

and of

Outer multiplication is a non-commutative, associative operation. The type 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 array, and a row vector, i.e. a type array, gives a matrix, i.e. a type 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
There are 3 references to this entry.

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 MSC:

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

Pending Errata and Addenda

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