outer multiplication
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 refers
to the vector space of type tensor arrays, i.e. maps
where is some finite list of index labels, and where is a field.
Let be natural numbers. Outer multiplication is a
bilinear operation
that combines a type tensor array and a type tensor array to produce a type tensor array (also written as ), defined by
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
; they commute with everything. Thus, we can embed into
the direct sum
and thereby endow the latter
with the structure of an -algebra
11We 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 in denoting outer
multiplication. These topics are dealt with in the entry pertaining
to abstract tensor algebra..
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:
Title | outer multiplication |
---|---|
Canonical name | OuterMultiplication |
Date of creation | 2013-03-22 12:40:31 |
Last modified on | 2013-03-22 12:40:31 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 4 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 15A69 |
Related topic | TensorProductClassical |
Related topic | TensorProduct |