Matrix Multiplication and sigma notation

[Bashyboy]

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?

 

[DavidHume]

"Sigma" notation is simply summation.

Matrix multiplication is the summation of the rows of one matrix multiplied by the columns of another matrix.

 

[Bashyboy]

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.

 

[D H]

The way you explain makes it seem that Wikipedia has defined it incorrectly.

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."

 

[DavidHume]

I think my explanation agrees with Wikipedia. Here's an example:

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

  A
1 3
5 7

   B
2 4
6 8

Matrix multiply (in the linear executable notation J - see jsoftware.com):

   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):

            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).

 

[Bashyboy]

I think I am beginning to understand it, now. I'll have to re-read the posts a few more times. Thank you, all.