Matrix Multiplication & Splitting

Hi I'm trying to extract given values from a large matrix and mulitply it by other known matrices e.g.

A = [1 2 3: 4 5 6: 7 8 9] % 3x3 Matrix which is always the same
B = [1 2 3] % 3x1 Matrix which also stays the same

C = % A matrix of 3x12 or 3x15 etc, e.g. The matrix could be split into indiviudal 3x3 matrices

As shown above I compute the matrix C which is formed from varying numbers of 3x3 matrices. I'm then attempting to call each 3x3 from the matrix and multiply it by the A and B e.g.

D1 = A * B * C(:,1:3)
D2 = A * B * C(:,4:6)
D3 = A * B * C(:,7:9) % etc until I have done it for all the sub 3x3 matrices

As shown the large matrix C is split into equal 3x3 matrices and then is multiplied by A and B. I was wondering how this could be done for C being any 3 by matrix e.g. 3x12 or 3x36 i.e. it can be evenly split into individual 3x3 matrices.

Any help is greatly appreciated, thanks!

[Fangjun Jiang]

?? B = [1 2 3] % 3x1 Matrix which also stays the same

either way, for the first 3x3 part of C, A*B*C is invalid

[Bruno Luong]

"B = [1 2 3] % 3x1 Matrix which also stays the same"

Make it (1 x 3)

[Rahul Marwaha]

Sorry to add clarification the A (3x3) should multiply C(3x3) to form a new 3x3 matrix. That matrix (3x3) can then be multiplied by B (3x1) to produce a new 3x1. Hope this clarifys the issue that a 3x3 should be found to then find the 3x1.

[Steven Lord]

If you adopt the approach suggested by the other posters and turn your C matrix into a 3-dimensional array D, in release R2020b and later you could use the pagemtimes function to compute the product of your A matrix with each page of D.

Answer 1

[Ive J]

Given your input criteria, you can simply reshape C:

sz = size(C, 2);
C = reshape(C, 3, 3, sz/3); 
D = arrayfun(@(i)A*B*C(:,:,i), 1:sz/3, 'UniformOutput', false);

[Rahul Marwaha]

Hi, thanks for answering! I have tried this method but I get the following:

Error using arrayfun

All of the input arguments must be of the same size and shape.

Previous inputs had size 12 in dimension 2. Input #3 has size 13

Not sure what is causing the error so any advice would be much appreciated!

[Ive J]

I guess you meant B*A*C, otherwise it's impossible!

A = [1 2 3; 4 5 6; 7 8 9];
B = [1 2 3];
C = reshape(1:36, 3, 12);

sz = size(C, 2);
C = reshape(C, 3, 3, sz/3); 
D = arrayfun(@(i)B*A*C(:,:,i), 1:sz/3, 'UniformOutput', false)
  1×4 cell array

    {1×3 double}    {1×3 double}    {1×3 double}    {1×3 double}
    

[Rahul Marwaha]

Thank you so much, I apologise it should have been as you've shown I totally forgot the order of multiplicaiton that would render the calculation impossible.

Answer 2

[James Tursa]

Based on your latest posts, it sounds like you really want A*C(3x3 slice)*B. So again it would be nice to do all the multiplies in one fell swoop and then pick off your desired results. E.g.,

D = B.' * reshape(permute(reshape(A*C,3,3,[]),[2 1 3]),3,[]);
D = reshape(D,3,[]);

and the results are in D(:,1), D(:,2), etc.

Or just use the following as Steven Lord suggests:

D = pagemtimes(reshape(A*C,3,3,[]),B);

Answer 3

[James Tursa]

Your dimensions don't work. A*B is going to be 3x1. You can't multiply this by a 3xN matrix.

That being said, suppose you did have dimensions that did work for A*B. Then you don't need to split anything up for the multiply. You can just do A*B*C directly and then pick off the columns of the result that you want. E.g., suppose we have the simpler problem below

A is 3x3

C is 3x12

and we want A multiplied by individual 3x3 parts of C. Then instead of doing these individual matrix multipies

D1 = A*C(:,1:3)

D2 = A*C(:,4:6)

D3 = A*C(:,7:9)

D4 = A*C(:,10:12)

You can just do this all in one fell swoop

D = A*C

and then reshape it for convenience

D = reshape(D,3,3,[])

Then you can pick off D(:,;1), D(:,:,2), etc. for your results.

[Rahul Marwaha]

Hi James, thanks for responding. It may not be clear but the 3xN matrix should be a 3x3 matrix. Essentially in my case C is a large 3xN matrix where N is any number divisible by 3 so that the matrix can be reshaped to form individual 3x3 matrices i.e C is 3x3 at minimum or 3x6 or 3x9 etc.

I'm then looking to have a 3x3 * 3x3 * 3x1 which would give a resultant 3x1 matrix. That is what i'm trying to resolve. Not sure if that helps but appreciate if you have any alternative suggestions, thanks.

[James Tursa]

If B is really 1x3 and the order is supposed to be B*A*C(:,slice), then it is just

D = B*A*C;

Then you can reshape

D = reshape(D,1,3,[]);

and you can pick off D(:,:,1), D(:,:,2), etc. for the results.