# "Element-wise" matrix vector multiplication in matlab

## Alec Jacobson

## February 23, 2011

### or

## Multiply each matrix in a list of matrices against corresponding vector in list of vectors

Here's a code snippet to take a stacked list of m by n matrices, S, and multiply each m by n matrix by a corresponding n-length row vector in a list of row vectors, V. Resulting in a list of m-length row vectors:
First of all, make your *vertical* stack of matrices S. So that S(i:(i+m-1),:) = Mi and so forth. And be sure that your list of row vectors is s by n, in V, where s is the number of matrices and vectors.
Then you may issue:
```
I = repmat([1:(m*s)],1,n);
J = repmat(reshape(repmat(1:n:(s*n),m,1),1,s*m),1,n)+reshape(repmat(0:(n-1),s*m,1),1,s*m*n);
SV = reshape(sparse(I,J,S(:))*reshape(V',s*n,1),m,s)';
```

Often I find that I also have a bunch of matrices stored in a 3d-array, say, T. Where T(:,:,i) = Mi and so forth. I can easily convert T, an m by n by s 3d-array into S, an m*s by n array, using:
```
S = reshape(permute(T,[2,1,3]),[n,s*m])';
```

**Update:**
Here's working code that uses the above:
```
s = 4;
m = 2;
n = 3;
% generate some bogus matrices
S = repmat(eye(m,n),s,1).*repmat(reshape(repmat(1:s,m,1),s*m,1),1,n);
% generate some bogus row vectors
V = reshape(1:s*n,n,s)';
% compute "element-wise" matrix-vector multiplication
I = repmat([1:(m*s)],1,n);
J = repmat(reshape(repmat(1:n:(s*n),m,1),1,s*m),1,n)+reshape(repmat(0:(n-1),s*m,1),1,s*m*n);
SV = reshape(sparse(I,J,S(:))*reshape(V',s*n,1),m,s)';
% view result
SV
% convert vertical stack into 3d array stack
T = permute(reshape(S,[m,s,n]),[1,3,2]);
% convert 3D array stack to vertical stack
TS = reshape(permute(T,[2,1,3]),[n,s*m])';
```

**Update:** Here's the **for loop** version, which is much more straight forward, but starts to slow down in comparison for large s:
```
SV = zeros(s,m);
for ii = 0:(s-1)
SV(ii+1,:) = (S((m*ii+1):(m*ii+m),:)*(V(ii+1,:)'))';
end
```