Array vs. Matrix Operations

A row vector and a column vector have compatible sizes. If you add a 1-by-3 vector to a 2-by-1 vector, then each vector implicitly expands into a 2-by-3 matrix before MATLAB executes the element-wise addition.

x = [1 2 3]
x =
     1     2     3
y = [10; 15]
y =
    10
    15
x + y
ans =
    11    12    13
    16    17    18

If the sizes of the two operands are incompatible, then you get an error.

A = [8 1 6; 3 5 7; 4 9 2]
A =
     8     1     6
     3     5     7
     4     9     2
m = [2 4]
m =
     2     4
A - m

Matrix dimensions must agree.

This is from the MATLAB "Array vs. Matrix Operations page". Why does the second example output an error while the first doesn't? I see that the second example says that "matrix dimensions must agree", but why did that error not occur for the first example? A further explanation of this would be great. Thank you!

[Stephen23]

Note that your title "Array vs. Matrix Operations" actually refers to different kinds of operators, not specifically to compatible array sizes for basic array operations:

https://www.mathworks.com/help/matlab/matlab_prog/array-vs-matrix-operations.html

Answer 1

[Stephen23]

What "compatible sizes" means is explained on this page:

https://www.mathworks.com/help/matlab/matlab_prog/compatible-array-sizes-for-basic-operations.html

I will not copy the entire page here, but the main points are:

• scalar dimensions can be expanded/contracted to match the other array.
• non-scalar dimensions must have exactly the same size.

That is all. So your first example works because (note the scalar dimensions):

• 1x3 can be expanded to 2x3
• 2x1 can be expanded to 2x3

But your second example fails because

• 1x2 can be expanded to 3x2
• 1x2 cannot be expanded/contracted to match 3x3,nor can 3x3 be expanded/contracted to match 1x2, because in the second dimension neither is scalar, nor do they have the same size.