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 $\mathrm{T}^{p,q}$ refers to the vector space of type $(p,q)$ tensor arrays, i.e. maps

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

\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 $X Y$ (also written as $X\otimes Y$ ), defined by

(XY)^{i_{1}\ldots i_{p_{1}}i_{p_{1}+1}\ldots i_{p_{1}+p_{2}}}_{j_{1}\ldots j_{% q_{1}}j_{q_{1}+1}\ldots j_{q_{1}+q_{2}}}=X^{i_{1}\ldots i_{p_{1}}}_{j_{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, $X Y$ . Quite obviously then, the size of $X Y$ is the size of $X$ times the size of $Y$ , and the index slots of the product $X Y$ 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

\bigoplus_{p,q\in\mathbb{N}}\mathrm{T}^{p,q},

and thereby endow the latter with the structure of an $\mathbb{K}$ -algebra¹¹We 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..

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:

\begin{pmatrix}a\\ b\\ c\end{pmatrix}\otimes\begin{pmatrix}x&y&z\end{pmatrix}=\begin{pmatrix}ax&ay&az% \\ bx&by&bz\\ cx&cy&cz\end{pmatrix},\quad a,b,c,x,y,z\in\mathbb{K}

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