1. ## Direct Product

Given two matrices $\displaystyle A=\{A_{ij}\}_{n \times m} , B=\{B_{kl}\}_{P \times Q}$, how do i define the direct product of $\displaystyle A \otimes B$?

I understand what it means, $\displaystyle (A \otimes B)_{ij} = A_{ij}B$ but how do i define it using the correct notation?
Thanks

2. Actually, you don't define the "direct product" of two matrices, that's the wrong word. The direct product is a product of sets, not matrices. What youj are talking about, $\displaystyle A\otimes B$, is the "outer product" or "tensor product". It is an array of all possible products of numbers in A times numbers in B "organized" in four-dimensional array. If A is an n by m matrix then it has mn numbers in it in n rows and m columns. If B is an P by Q matrix then it has PQ numbers in it in P rows and Q columns. $\displaystyle A\otimes B$ would have mnPQ members arranged along 4 axes, m numbers along one axis, n along another, P along the third axis, and Q along the fourth. Of course, that would be very hard to write in three dimensional space, not to mention a two dimensional page!

What you might do is this: write the first (ij= 11) number in B times each member of A as an m by n matrix. Write the next number in the first row (ij= 12) of B times each member of A as another m by n matrix immediately below it. Do that for each of the Q members of the first row of B. Then, on a new page repeat with the numbers on the second row of B. Do that, using a new page, for each of the P rows of B. You will now have P pages, each containing Q n by m matrices, giving a full representation of $\displaystyle A\otimes B$.

For example, if m= n= P= Q= 2, so that A and B are 2 by two matrices, say
$\displaystyle A= \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ and $\displaystyle B= \begin{bmatrix}2 & 1 \\ 0 & 1\end{bmatrix}$
then the first page would have the two matrices
$\displaystyle \begin{bmatrix}2 & 4 \\ 6 & 4\end{bmatrix}$
(The "2" in the B matrix times A) and
$\displaystyle \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$
(the "1" in the B matrix times A) while the second page would have
$\displaystyle \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$
and
$\displaystyle \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$
showing all 2 x 2 x 2 x 2= 16 numbers in $\displaystyle A\otimes B$ in a "2 by 2 by 2 by 2" array.

3. Originally Posted by HallsofIvy
Actually, you don't define the "direct product" of two matrices, that's the wrong word. The direct product is a product of [b]sets[b], not matrices. What youj are talking about, $\displaystyle A\otimes B$, is the "outer product" or "tensor product".
This is also commonly called the Kronecker product.