creating smaller matrix from a large matrix

I want to create a 16*16 square matrix from 4*4 square matric based on a condition that 4*4 (i.e. 256) elements add together to form a new element for smaller matrix. Example is shown below.

[Torsten]

If this is an IQ test, I didn't pass it...

[Manoj Kumar V]

All the 16 elements in the blue boxes were added to get a number 72. And all the 16 elements in yellow boxes were added to get a number 92. Likewise, I want to add all 4*4 (16 elements) to form a new smaller 4*4 matrix in matlab. It is about contracting the dimensions of a matrix.

Answer 1

[Voss]

M = randi(10,16,16); % 16-by-16 matrix of integers between 1 and 10

disp(M);

     7     4     5     5     2     3     4    10    10     5     1     8     8     4     1     4
     3     5     6     7    10     6     6     8     7     5     3     2    10     5     6     4
     7     4     9     6     1     9     8     7     2     8     3     8     2     7     4     5
     3    10     6     7     9     7     6     1     9     3     7     5     3     2     7     5
     8     8     7     7     2     7     4    10     4     3     5     5     6     8     1     5
     4     9     2     4     9     3     9    10    10     7     6     3     7     8     3    10
     9     7     1     2    10     5     4     6     5     6     9     2     9    10     4    10
    10     3     7     2     6     4    10     3     5     6     2     6     2     5     5     1
     7     6     7     8     7     8     2     2     5     8     7     2     5     4    10     2
    10     9    10     9     6     5    10     9     5     2     8     6     3     8     8     1
    10     7     8     2     6     9     9     3     5     2    10     3     4     8     1     4
     1    10     2     8     8     8     8     2     4     2     7     5     9    10    10     2
     1     9     1     3     3     7     3     1     8     5     4     3     4     1     1     6
     7    10     3     6     2     5     3     8     3     1     1     4     1     1     1     7
     3     4     3     2     9     2     8     9     4     2    10     8     6     8     8     1
     5     6     7     4     3     1     4     8     3     6    10     3     6     5     6    10

siz = [4 4]; % size of smaller matrices to divide M into

args = arrayfun(@(n,s)n*ones(1,s/n),siz,size(M),'UniformOutput',false);

m = cellfun(@(b)sum(b,'all'),mat2cell(M,args{:}))

m = 4×4

94 97 86 77 90 102 84 94 114 102 81 89 74 76 75 72

% check
m(1,1) == sum(M(1:4,1:4),'all')

ans = logical
   1

m(2,1) == sum(M(5:8,1:4),'all')

ans = logical
   1

m(3,4) == sum(M(9:12,13:16),'all')

ans = logical
   1

[Voss]

Or using some loops:

M = randi(10,16,16); % 16-by-16 matrix of integers between 1 and 10
disp(M)

     9     2     2     5     4     1     5     9     5     5     1    10     3     3     3     3
     1     7     6    10     4     2     4     3     7     3     1     3     6     3     7     9
    10     2     9     8     8     8     1     8     1     7     2     7     3     6     3     1
     3     8     6     2     4     8     1    10     5    10     7     4     8     7     5    10
     8     9     5     4    10    10     8     2     7     3     6     9     2     2     4     8
     3     2     9     3     7    10     7     4     3     6     8     8     2     4     2     7
     8    10     4     5     9     8     5     4     2     5     1     4     5     4     9     3
     9    10     3     7     1     5     4     5     1     1     7     1     9     5     7     7
     9     1    10     8     3     7     6     6     8     1     9     1     9     1     8     5
     4     5     3     6     6     9     4     8     1    10     2     9     7    10     7     8
     7     2     9     9     5     1    10     1     7     3     9     2     4     1     4     2
    10     5     6     9     5     7     3     8     3     9     8     8     3     8     7     3
     5     4     9     2     4     5     6     1     7     1     9     1     3     5     8     9
     7     8     8     4     4     5     4     4     8     3     7     7     5     4    10    10
     5     7     5     4     1     7     7    10     7     9    10     4    10     9     8     6
     3     6     7     4     8     5     1     2     6     8     6     2     1     5     7     3

siz = [4 4]; % size of smaller matrices to divide M into

n = size(M)./siz;
m = zeros(n);
for ii = 1:n(1)
    for jj = 1:n(2)
        m(ii,jj) = sum(M((ii-1)*siz(1)+(1:siz(1)),(jj-1)*siz(2)+(1:siz(2))),'all');
    end
end

disp(m)

    90    80    78    80
    99    99    72    80
   103    89    90    87
    88    74    95   103

% check
m(1,1) == sum(M(1:4,1:4),'all')

ans = logical
   1

m(2,1) == sum(M(5:8,1:4),'all')

ans = logical
   1

m(3,4) == sum(M(9:12,13:16),'all')

ans = logical
   1

Answer 2

[Torsten]

You mean

rng("default")
A = rand(16);
for i = 1:4
    for j = 1:4
        B = A((i-1)*4+1:i*4,(j-1)*4+1:j*4);
        A_compressed(i,j) = sum(B(:));
    end
end
A_compressed

A_compressed = 4×4

9.5135 8.3613 7.5381 7.0071 7.4615 8.0673 5.5216 8.6224 10.1127 6.9479 8.0587 7.2866 7.9292 8.0528 7.6726 7.9785

?

[Manoj Kumar V]

Thank You Torsten. There seems to be numbers with decimal points for the answer, which shall not be expected. Voss has explained with the example.

[Torsten]

Well, you didn't mention the condition that the matrix elements should be integers - so I generated doubles for the 16x16 matrix. But no problem ...

Answer 3

[Adam Danz]

Here are two more options. Neither are very readable but I knew there was a grouping solution and a vectorized solution and I was in the mood for a challenge.

Grouping solution

The variable group is a matrix the same size as A that groups the values of A into 4x4 sub-matrixes.

rng("default")
A = randi(10,16);

group = repelem(reshape(1:16,4,4)',4,4);
sumVec = groupsummary(A(:),group(:),'sum');  % alternative: =splitapply(@sum,A(:),group(:));
m = reshape(sumVec,4,4)'

m = 4×4

104 92 85 80 82 88 62 94 107 78 87 83 87 87 86 89

Vectorized solution

This reshapes matrix A into a 4-dimensional array and then permutes the array so that sum can be applied acros the first two dimensions. Then the results are reshaped back into a matrix.

m = reshape(sum(permute(reshape(A',4,4,4,4),[1 3 2 4]),[1,2]),4,4)'

m = 4×4

104 92 85 80 82 88 62 94 107 78 87 83 87 87 86 89