outer multiplication

Note: the present entry employs the terminology and notation definedand described in the entry on tensor arrays. To keep thingsreasonably self contained we mention that the symbol ${\mathrm{T}}^{p,q}$ refersto the vector space of type $(p,q)$ tensor arrays, i.e. maps

$$I^p\times I^q\to 𝕂,$$

where $I$ is some finite list ofindex labels, and where $𝕂$ is a field.

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

$${\mathrm{T}}^{p_1,q_1}\times {\mathrm{T}}^{p_2,q_2}\to {\mathrm{T}}^{p_1+p_2,q_1+q_2}$$

that combines a type $(p_1,q_1)$tensor array $X$ and a type $(p_2,q_2)$ tensor array $Y$ toproduce a type $(p_1+p_2,q_1+q_2)$ tensor array $XY$ (alsowritten as $X\otimes Y$), defined by

$${(XY)}_{j_1\mathrm{\dots }{j}_{q_1}{j}_{q_1+1}\mathrm{\dots }{j}_{q_1+q_2}}^{i_1\mathrm{\dots }{i}_{p_1}{i}_{p_1+1}\mathrm{\dots }{i}_{p_1+p_2}}=X_{j_1\mathrm{\dots }{j}_{q_1}}^{i_1\mathrm{\dots }{i}_{p_1}}{Y}_{j_{q_1+1}\mathrm{\dots }{j}_{q_1+q_2}}^{i_{p_1+1}\mathrm{\dots }{i}_{p_1+p_2}}$$

Speaking informally, what is going on above is that we multiplyevery 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 $XY$ isthe size of $X$ times the size of $Y$, and the index slots of theproduct $XY$ are just the union of the index slots of $X$ and of $Y$.

Outer multiplication is a non-commutative, associative operation. Thetype $(0,0)$ arrays are the scalars, i.e. elements of$𝕂$; they commute with everything. Thus, we can embed $𝕂$ intothe direct sum

$$\underset{p,q\in \mathbb{N}}{\oplus }{\mathrm{T}}^{p,q},$$

and thereby endow the latterwith the structure of an $𝕂$-algebra.

By way of illustration we mention that the outer product of a columnvector, 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: