# 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

$${I}^{p}\times {I}^{q}\to \mathbb{K},$$ |

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

$${\mathrm{T}}^{{p}_{1},{q}_{1}}\times {\mathrm{T}}^{{p}_{2},{q}_{2}}\to {\mathrm{T}}^{{p}_{1}+{p}_{2},{q}_{1}+{q}_{2}}$$ |

that combines a type $({p}_{1},{q}_{1})$ tensor array $X$ and a type $({p}_{2},{q}_{2})$ tensor array $Y$ to produce a type $({p}_{1}+{p}_{2},{q}_{1}+{q}_{2})$ tensor array $XY$ (also written 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 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
product^{} $XY$ 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^{}

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

and thereby endow the latter
with the structure^{} of an $\mathbb{K}$-algebra^{}^{1}^{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..

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:

$$\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \\ \hfill c\hfill \end{array}\right)\otimes \left(\begin{array}{ccc}\hfill x\hfill & \hfill y\hfill & \hfill z\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill ax\hfill & \hfill ay\hfill & \hfill az\hfill \\ \hfill bx\hfill & \hfill by\hfill & \hfill bz\hfill \\ \hfill cx\hfill & \hfill cy\hfill & \hfill cz\hfill \end{array}\right),a,b,c,x,y,z\in \mathbb{K}$$ |

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 |