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 Tp,q refers to the vector spaceMathworldPlanetmath of type (p,q) tensor arrays, i.e. maps


where I is some finite list of index labels, and where 𝕂 is a field.

Let p1,p2,q1,q2 be natural numbersMathworldPlanetmath. Outer multiplication is a bilinear operation


that combines a type (p1,q1) tensor array X and a type (p2,q2) tensor array Y to produce a type (p1+p2,q1+q2) tensor array XY (also written as XY), defined by


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 productPlanetmathPlanetmathPlanetmath XY are just the union of the index slots of X and of Y.

Outer multiplication is a non-commutative, associative operationMathworldPlanetmath. The type (0,0) arrays are the scalars, i.e. elements of 𝕂; they commute with everything. Thus, we can embed 𝕂 into the direct sumMathworldPlanetmathPlanetmathPlanetmathPlanetmath


and thereby endow the latter with the structureMathworldPlanetmath of an 𝕂-algebraPlanetmathPlanetmath11We 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 productPlanetmathPlanetmathPlanetmath 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 vectorMathworldPlanetmath, 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:

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