Element wise multiplication of vectors within matrices when multiplying matrices.

Hi there, I wonder if you can help!....

I would like to multiply multiple 4x4 matrices together but each element of each matrix is a variable, and so in MATLAB is a vector. I therefore want to perform matrix multiplication on the matrices but element by element multiplication of the vectors inside, is this possible?

e.g when multiplying M1*M2 where M1=[A B ; C D], M2=[E F; G H] and A-H are all vectors

I want MATLAB to return [A.*E+B.*G A.*F+B.*H ; C.*E+D.*G C.*F+D.*H]

I can type out the formula for 2 matrices easily, but 10 matrices? It gets long very quickly! Can this be done quickly?

Many thanks, Luke

Answer 1

[AJ von Alt]

One way to perform the desired operation is to use a for loop to iterate through the third dimension of M1 and M2 and perform 2-D matrix multiplication at each level.

k = size( M1 , 3 ); %the size of the third dimension

for i = 1:k % for each level of the 3-D matrix
    OUT(:,:,i) = M1(:,:,i) * M2(:,:,i);  % multiply the corresponding 2D matrices
end

Assuming your vectors are all of length k with M1 of size ( m X n X k ) and M2 size ( n X p X k ), the result of this operation OUT will be size ( m X p X k ).

Below is a complete demonstration of this operation.

%%Inputs
m = 2;
n = 3;
p = 4;
M1(:,:,1) = 1 * ones( m , n );
M1(:,:,2) = 2 * ones( m , n );
M1(:,:,3) = 3 * ones( m , n );

M2(:,:,1) = 1 * ones( n , p );
M2(:,:,2) = 2 * ones( n , p );
M2(:,:,3) = 3 * ones( n , p );

%%Perform 2-D matrix multiplication at each level of the 3-D matrix

k = size( M1 , 3 ); %the size of the third dimension

OUT = zeros( m , p , k ); % preallocate OUT

for i = 1:k % for each level of the 3-D matrix
    OUT(:,:,i) = M1(:,:,i) * M2(:,:,i);  %multiply the corresponding 2D matrices
end

[Luke]

Thank you for your help, I will try this!