Multiplying Matrices and Identifying Dimensionality

Date: 07/16/2003 at 13:03:39
From: Rick
Subject: Multiplying Matrices and Identifying Dimensionality

I understand adding and subtracting matrices, but not multiplying 
them. I have this question, how do I do this?

[-2  4  -6]    [1  -1  1]
[ 1  -1  0]    [3  0  -2]
[ 0   2  5]    [4  -6  7]

Also, how can I find the dimensionality of the following matrix?

[2]
[4]  [3  1  7]
[8]

would 3x1  and 1x3 be correct and therefore they cannot be multiplied 
because the inner dimensionalities are different?

Thank you very much,
Rick

Date: 07/16/2003 at 23:11:39
From: Doctor Peterson
Subject: Re: Multiplying Matrices and Identifying Dimensionality

Hi, Rick.

Your second example is a 3x1 times a 1x3, so the inner dimensions ARE 
the same (both 1), and they CAN be multiplied. Just take each row of 
the first and multiply it by each column of the second; in this case 
each is just a single pair of numbers:

       [ 3   1   7 ]

  [2]  [2*3 2*1 2*7]   [ 6  2 14]
  [4]  [4*3 4*1 4*7] = [12  4 28]
  [8]  [8*3 8*1 8*7]   [24  8 56]

Do you see how I did it? In this case, it looks like a multiplication 
table, each row and column of the product being the product of the 
number in that row of the first matrix and the number in that column 
of the second matrix, which I wrote above for clarity.

To multiply matrices with more than one column, you have to multiply 
and add, rather than just multiply. For a simple example, look at

  [1 2] [5 3]
  [3 4] [2 4]

I can write them as above,

         [   5       3   ]
         [   2       4   ]

  [1 2]  [1*5+2*2 1*3+2*4]   [ 9 11]
  [3 4]  [3*5+4*2 3*3+4*4] = [23 25]

Here I "multiplied" the row [1 2] by the column [5,2] by multiplying 
corresponding pairs and adding the products: 1*5 + 2*2. Repeat that 
for all pairs of row and column, and you're done. Note that the inner 
dimensions have to agree, or you can't pair up numbers to multiply.

Here is a site that covers matrices nicely, which may help you when 
your book isn't clear:

   S.O.S. Math - Matrix Algebra
   http://www.sosmath.com/matrix/matrix.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/17/2003 at 09:26:47
From: Rick
Subject: Multiplying Matrices and Identifying Dimensionality

This makes sense now, but one more question. Would I do this one below 
the same way as my second one I showed you last time? 
    
           [1]
           [5]
[2 5 9]    [4]

Would it be....

2*1  2*5   2*4
5*1  5*5   5*4
9*1  9*5   9*4
 
Would this above be the correct way to do it? And then multiply the 
numbers of course.

Thanks.

Date: 07/17/2003 at 09:36:53
From: Doctor Peterson
Subject: Re: Multiplying Matrices and Identifying Dimensionality

Hi, Rick.

You multiplied in the wrong order. (Matrix multiplication is NOT 
commutative.) When you multiply a 1x3 and a 3x1, you get a 1x1; you 
multiplied the 3x1 and the 1x3 and got a 3x3.

Here is the correct work:

   +-----------[1]
   | +---------[5]
   | | +-------[4]
   | | |
  [2 5 9] [2*1+5*5+9*4] = [63]

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/