Given an -by- matrix with entries , we get an -by- matrix whose th entry is , i.e.
10. Dot product, 2
You may have noticed that the definition of the dot product looks a lot like matrix multiplication. In fact, it is a special case of matrix multiplication: Technically, the matrix product gives a 1-by-1 matrix whose unique entry is the dot product, but let's not be too pedantic.
Here, we took the column vector and turned it on its side to get a row vector which we call the transpose of , written: .
More generally, you can transpose a matrix:
So the rows of become the columns of .
With all this in place, we observe that the dot product is .
Writing out the th entry of using index notation, we get: Similarly expanding we get The two expressions differ only by the order of the factors and .
The order of these factors doesn't matter: and are just numbers (entries of and ), so they commute. This is one reason index notation is so convenient: it converts expressions involving noncommuting objects like matrices into expressions involving commuting quantities (numbers).