5 Ways to Look at Matrix Multiplication
Hello, I will explain 5 interpretations of matrix multiplication which are necessary to know for machine learning and data science.
A reminder about matrix notation:
We are considering the following general case for all 5 parts.
The first way is actually the definition of matrix multiplication.
- Each item of matrix C is dot product of the corresponding row from matrix A and corresponding column from matrix B.
Reminder of dot product:
From now on all the followings are just a different interpretation of the first.
2. Each column of C is a linear combination of columns of A with values of the corresponding column in B as weights.
I have written it down in images 5 and 6:
Let’s take a look at first column of C with perspective of image 5:
3. Matrix C can be calculated with adding n matrices of the same size as C when matrix i is multiplication of column i of A and row i of B.
Let’s clarify it with an example:
4. Each row of C is the corresponding row in A multiplied by matrix B.
5. We can divide matrices A and B into blocks and calculate C this way:
The way we divide A and B into blocks might have different options but we should consider that the dimensions of the resulting blocks match the rules of matrix multiplication.
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