What does it even mean to add a 1x3 matrix to a 3x1 matrix ?

▲ freehorse a day ago | parent | next [-]

This is about how array operations in matlab work. In matlab, you can write things such as

    >> [1 2 3] + 1
    ans = [2 3 4]

In this case, the operation `+ 1` is applied in all columns of the array. In this exact manner, when you add a (1 x m) row and a (n x 1) column vector, you add the column to each row element (or you can view it the other way around). So the result is as if you repeat your (n x 1) column m times horizontally, giving you a (n x m) matrix, do the same for the row vertically n times giving you another (n x m) matrix, and then you add these two matrices. So basically adding a row and a column is essentially a shortcut for repeating adding these two (n x m) matrices (and runs faster than actually creating these matrices). This gives a matrix where each column is the old column plus the row element for that row index. For example

    >> [1 2 3] + [1; 2; 3]
    ans = [2 3 4
           3 4 5
           4 5 6]

A very practical example is, as I mentioned, getting all differences between the elements of a time series by writing `S - S'`. Another example, `(1:6)+(1:6)'` gives you the sums for all possible combinations when rolling 2 6-sided dice.

This does not work only with addition and subtraction, but with dot-product and other functions as well. You can do this across arbitrary dimensions, as long as your input matrices non-unit dimensions do not overlap.

▲ quietbritishjim 20 hours ago | parent | prev [-]

It means the same thing in MATLAB and numpy:

   Z = np.array([[1,2,3]])
   W = Z + Z.T
   print(W)

Gives:

   [[2 3 4]
    [3 4 5]
    [4 5 6]]

It's called broadcasting [1]. I'm not a fan of MATLAB, but this is an odd criticism.

[1] https://numpy.org/devdocs/user/basics.broadcasting.html#gene...

	▲	adgjlsfhk1 20 hours ago \| parent [-]
		One of the really nice things Julia does is make broadcasting explicit. The way you would write this in Julia is `Z = [1,2,3] W = Z .+ Z' # note the . before the + that makes this a broadcasted` This has 2 big advantages. Firstly, it means that users get errors when the shapes of things aren't what they expected. A DimmensionMismatch error is a lot easier to debug than a silently wrong result. Secondly, it means that julia can use `exp(M)` etc to be a matrix exponential, while the element-wise exponential is `exp.(M)`. This allows a lot of code to naturally work generically over both arrays and scalars (e.g. exp of a complex number will work correctly if written as a 2x2 matrix)