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Matrix Multiplication and sigma notation

  1. Oct 17, 2012 #1
    Hello,

    I have read several different sources on this very topic, and the one thing that confused a little was defining it using sigma notation. Could some please explain to be what it means?
     
  2. jcsd
  3. Oct 17, 2012 #2
    "Sigma" notation is simply summation.

    Matrix multiplication is the summation of the rows of one matrix multiplied by the columns of another matrix.
     
  4. Oct 18, 2012 #3
    The way you explain makes it seem that Wikipedia has defined it incorrectly.

    http://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29

    The one thing that I don't quite understand about their sigma definition is, how does does i and j run through their values? I can see that k runs from 1 element to m elements, which would be the elements in the column for A, and the elements in the row for B.
     
    Last edited: Oct 18, 2012
  5. Oct 18, 2012 #4

    D H

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    Staff Emeritus
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    That's exactly how Wikipedia defines it. Look at the picture: A row times a column. Or read the text regarding the calculation of (AB)ij: "Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also their vector dot product."
     
  6. Oct 18, 2012 #5
    I think my explanation agrees with Wikipedia. Here's an example:

    Say we have two tables "A" and "B":

    Code (Text):
      A
    1 3
    5 7

       B
    2 4
    6 8
    Matrix multiply (in the linear executable notation J - see jsoftware.com):

    Code (Text):
       A+/ . *B
    20 28
    52 76
    This is illustrated below by positioning B above and to the right with A down and to the left to highlight that we
    1) multiply the columns of B by the rows of A and
    2) sum those products (add them together)

    Here's my attempt to illustrate this procedure (assuming multiplication before addition):
    Code (Text):

                2          4
                      6          8
    1   3     1*2 + 3*6  1*4 + 3*8  =   20  28          
    5   7     5*2 + 7*6  5*4 + 7*8  =   52  76        
    I've also staggered the rows of B to align them with the relevant portion of the cross-product (and tagged it as "code" when it isn't in order to preserve the spacing).
     
  7. Oct 18, 2012 #6
    I think I am beginning to understand it, now. I'll have to re-read the posts a few more times. Thank you, all.
     
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