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Overview of matrix multiplication

1. Matrix Multiplication

As you've probably realized, matrix operations are fundamental to the ALS algorithm. We're going to review matrix multiplication and matrix factorization here. Let's start with multiplication.

2. Matrix Multiplication

Here we have two square matrices. In order to multiply them together, we make specific pairs of the values from the two matrices, and add the products of those pairs. We start at the top left-hand corner of each matrix, and create pairs moving to the right on the first matrix, and moving down on the second matrix one at a time. Each pair is multiplied, and the products from all pairs are added together. The final sum will make up one number of the resulting matrix. That's a lot to digest, so let's walk through an example. Starting at the top

3. Matrix Multiplication

left number of each matrix we have a pair of numbers, 1 and 9. We will multiply those numbers together. Then moving to the right on the first matrix, and down on the second matrix we have

4. Matrix Multiplication

2 and 6, then moving

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right again on the first matrix and down again on the second matrix we have 3 and 3. We have completed the first set of pairs, so let's add their products together. 1 times 9, plus 2 times 6 plus 3 times 3 is

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9 plus 12 plus 9, which gives us

7. Matrix Multiplication

30. 30 is the first number in our final matrix. From here we stay on the first row of the first matrix, but move on to the second column of the

8. Matrix Multiplication

second matrix. These pairs give us 1 and 8,

9. Matrix Multiplication

2 and 5,

10. Matrix Multiplication

and 3 and 2. 1 times 8 plus 2 times 5 plus 3 times 2 is equal to 8 plus 10 plus 6, which is

11. Matrix Multiplication

24. Moving to the next set of pairs, we multiply

12. Matrix Multiplication

1 and 7, 2 and 4, and 3 and 1. Their products are 7, 8 and 3 which makes

13. Matrix Multiplication

18. Once we've multiplied the first row of the first matrix by all columns of the second matrix, we then go through the same process for

14. Matrix Multiplication

the second row of the first matrix with all the columns of the second matrix,

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and so on

16. Matrix Multiplication

until all rows of the

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first matrix have been multiplied

18. Matrix Multiplication

by all columns of the second matrix. In this example, we multiplied two square matrices of the same dimensions. In reality, you can multiply any two matrices as long as the

19. Matrix Multiplication

number of columns of the first matrix matches the number of rows of the second matrix,

20. Matrix Multiplication

If they don't, then some values in one of the matrixces won't be paired, and multiplication can't be completed.

21. Let's practice!

Let's look at some examples, and practice matrix multiplication.