Is there any matlab documentation that can explain why multiplying empty arrays gives zero matrices?

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fa wu 2023-8-29

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评论： Steven Lord 2023-8-30

采纳的回答： Bruno Luong

在 MATLAB Online 中打开

Is there any matlab documentation that can explain why multiplying empty arrays gives zero matrices?

Here is the sample

A=double.empty(5,0);
B=double.empty(0,5);
C=A*B
C =
0     0     0     0     0
0     0     0     0     0
0     0     0     0     0
0     0     0     0     0
0     0     0     0     0

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Bruno Luong 2023-8-29

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编辑：Bruno Luong 2023-8-29

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From group theory point of view, sum/product (group operator) of an empty set gives the group neutral element. The most basic examples are

sum([])
ans = 0
prod([])
ans = 1

From that property, mathematically by definition of matrix multiplication; the product AB :=A*B of

A = double.empty(m,0)
B = double.empty(0,n)

is equal to

zeros(m,n)

since each element AB(i,j) of A*B is a sum of an empty set.

This is mathematically defined, not MATLAB own convention, no doc description should be necessary.

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Steven Lord 2023-8-29

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The product of an m-by-k matrix and a k-by-n matrix is an m-by-n matrix. This holds no matter what m, k, and n are. This is standard matrix multiplication.

In your example, m is 5, k is 0, and n is 5. Therefore by this general rule the result of multiplying a 5-by-0 and a 0-by-5 must be a 5-by-5 matrix.

So what should the elements of the resulting matrix be? There are a couple reasonable possibilities IMO. But if you write matrix multiplication naively (this is not how MATLAB implements it internally, BTW) you could do so with a double for loop using the dot product function dot.

A = zeros(5, 0);
B = zeros(0, 5);
result = naiveMatrixMultiply(A, B)
result = 5×5
     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     0

Why are the elements of the result 0? What's the dot product of a 1-by-0 row of A and a 0-by-1 column of B? By the (handwaving) definition of a dot product you multiply corresponding elements of the two vectors and sum all those products. But the empty row of A and column of B have no elements. What's the sum of no elements in MATLAB?

sum([])
ans = 0

Hopefully this makes sense.

function result = naiveMatrixMultiply(A, B)
assert(width(A) == height(B)) % number of columns of A must == number of rows of B
for whichRow = 1:height(A)
    for whichCol = 1:width(B)
        result(whichRow, whichCol) = dot(A(whichRow, :), B(:, whichCol));
    end
end
end

Stephen23 2023-8-29

编辑：Stephen23 2023-8-29

"I don't see the need for [A B], nor does [C; D],"

You missed the point: p is a variable that could be any positive integer, including zero. This means that in general you can write code using MATLAB that works correctly without any special cases, even for empty arrays. For example, imagine you want to filter and select some of your data (much like A,B,C,D) before doing some calculation: in general MATLAB can do this without any edge cases, even if the filtering returns no matching data (like B,D). The value of p is simply how many cases are returned by that filtering.

Imagine I want to split my data based on four states: if one of the states has no matches then I still want to plot the results of my calculation on all four states, not crash with an error like other (unhelpful, timewasting, inconsistent) programming languages.

This consistency is one of the very nice things about using MATLAB: it matches mathematics.

"Why empty array can be we “avoid program around edge cases”"

I suffer every time I have to write code in a language that does not have empty arrays: latent bugs that occur when suddenly a (e.g. filtered or imported) data set contains no elements. Solution: lots of special cases everywhere (thanks VBA, you've wasted a lot of my time due to this exact "feature").

Bruno Luong 2023-8-30

编辑：Bruno Luong 2023-8-30

a.' performs matrix transposition, if a is column vector as in the context it puts a as row vector

* is matrix multiplication.

So for a and b columns vectors of the same height a.'*b is sum(a.*b)

Actualy the right formula for dot replacement is a'*b not a.'*b as Paul wrote.

Steven Lord 2023-8-30

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The [] matrix in MATLAB is treated kind of specially by a lot of older functions, because as described in the post from Professor Nick Higham's blog that @Dyuman Joshi posted in an earlier comment, in the original version of MATLAB that was the empty matrix.

The behavior of sum of [] returning 0, if you're willing to allow one special quirk of concatenation related to the [] matrix (where it can be concatenated with a vector even though it doesn't have the same number of rows), can be shown as follows. If concatenating a vector and [] together results in that same vector (which it does):

a = 1:5
a = 1×5
     1     2     3     4     5
b = [a, []]
b = 1×5
     1     2     3     4     5

and if you accept that the sum of the concatenation of two arrays should be the same as the sum of each of those arrays added together:

sum12345 = sum(1:5) % 1:5 is the same as [1:3 4:5]
sum12345 = 15
sum123 = sum(1:3)
sum123 = 6
sum45 = sum(4:5)
sum45 = 9
sum12345 == sum123 + sum45 % true
ans = logical
   1

then sum([a []]) equals sum(a) + sum([]). But [a []] is just a, so that simplifies to sum(a) = sum(a) + sum([]). Subtract sum(a) from both sides and 0 = sum([]).

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