Matlab - Multiplying a matrix with every matrix of a 3d matrix

Question

I have two matlab questions that seem closely related.

1. I want to find the most efficient way (no loop?) to multiply a (A x A) matrix with every single matrix of a 3d matrix (A x A x N). Also, I would like to take the trace of each of those products. http://en.wikipedia.org/wiki/Matrix_multiplication#Frobenius_product

   This is the inner frobenius product. On the crappy code I have below I'm using its secondary definition which is more efficient.

2. I want to multiply each element of a vector (N x 1) with its "corresponding" matrix of a 3d matrix (A x A x N).

   function Y_returned = problem_1(X_matrix, weight_matrix)
   
   % X_matrix is the randn(50, 50, 2000) matrix
   % weight_matrix is the randn(50, 50) matrix
   
   [~, ~, number_of_matries] = size(X_matrix);
   Y_returned = zeros(number_of_matries, 1);
   for i = 1:number_of_matries
   %     Y_returned(i) = trace(X_matrix(:,:,i) * weight_matrix');
       temp1 = X_matrix(:,:,i)';
       temp2 = weight_matrix';
       Y_returned(i) =  temp1(:)' * temp2(:);
   end
   end
   
   
   function output = problem_2(vector, matrix)
   
   % matrix is the randn(50, 50, 2000) matrix
   % vector is the randn(2000, 1) vector
   
   [n1, n2, number_of_matries] = size(matrix);
   output = zeros(n1, n2, number_of_matries);
   for i = 1:number_of_matries
       output(:, :, i) = vector(i) .* matrix(:, :, i);
   end
   output = sum(output, 3);
   
   end
   

Answer 1

I assume you mean element-wise multiplication:

1. Use bsxfun:

   A = 10;
   N = 4;
   mat1 = randn(A,A);
   mat2 = randn(A,A,N);
   result = bsxfun(@times, mat1, mat2);
   

2. Use bsxfun with permute to align dimensions:

   A = 10;
   N = 4;
   vec1 = rand(N,1);
   mat2 = randn(A,A,N);
   result = bsxfun(@times, permute(vec1,[2 3 1]), mat2);