Speeding up some matrix arithmetic operation

Question

I have two matrices A and B.

B is just a matrix with only one of the diagonal elements non zero. All the non diagonal elements are zero as well. I have to calculate $A^{-1}B$. My $A^{-1}B$ matrix is sparse. In matlab I can do A\B. But is there any trick to further speed this up?

I have a bunch of B matrices in which only one of the diagonal elements are non zero and the non diagonal elements are zero. I cannot store the $A^{-1}$. Is there any way to speed this up?

Answer 1

Your problem can be simplified very much. Given that we want to store solution in a new matrix (C say), and

a) A is an NxN matrix

b) you know B only the main diagonal has elements in it (with itself being an NxN matrix)

c) you want A*B

C = zeros(size(A));
new_B = diag(B)';
[A_rows, A_cols] = size(A);
for i=1:A_rows
    C(i,:) = A(i,:).*new_B;
end

Answer 2

If the $(i,i)^{th}$ element of an otherwise zero matrix $B$ is equal to one, and you post-multiply some matrix $A^{-1}$ by this matrix, you are, in effect, extracting the $i^{th}$ column from $A^{-1}$ and setting all the other elements equal to zero. Since you need the full $i^{th}$ column of $A^{-1}$, you'll still have to invert $A$, but you don't need to actually do the post-multiplication; just create a zero matrix and replace the $i^{th}$ column with that of $A^{-1}$.