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Math Help - Direct Product

  1. #1
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    Direct Product

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

    I understand what it means, (A \otimes B)_{ij} = A_{ij}B but how do i define it using the correct notation?
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  2. #2
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    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, 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. 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 A\otimes B.

    For example, if m= n= P= Q= 2, so that A and B are 2 by two matrices, say
    A= \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} and B= \begin{bmatrix}2 & 1 \\ 0 & 1\end{bmatrix}
    then the first page would have the two matrices
    \begin{bmatrix}2 & 4 \\ 6 & 4\end{bmatrix}
    (The "2" in the B matrix times A) and
    \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}
    (the "1" in the B matrix times A) while the second page would have
    \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}
    and
    \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}
    showing all 2 x 2 x 2 x 2= 16 numbers in A\otimes B in a "2 by 2 by 2 by 2" array.
    Last edited by HallsofIvy; January 17th 2011 at 06:05 AM.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    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, A\otimes B, is the "outer product" or "tensor product".
    This is also commonly called the Kronecker product.
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