Solving linear matrix equation

Hi folks! I need some help (Though I have a doubt whether this is an appropriate question for this group).

Can you please help me to solve this equation: AB=A, where A is a known symmetric, singular matrix. And diagonal elements of B are also known.

For clarification: All elements of A is known. And only diagonal elements of B are known. As an example, you can consider: [1 -1/2 -1/2; -1/2 1 -1/2; -1/2 -1/2 1] and diag B=[ 3 3 3].

Thanks in advance.

[KSSV]

Show us A and B.

[Vladimir Sovkov]

Maybe, a kind of a general optimization procedure can be employed with the off-diagonal elements of B chosen as the variables to be optimized. If only the exact solution exists at all, which is not clear. I believe, our Optimizer package can help, see it at https://sourceforge.net/projects/optimizer-sovkov/.

[Sara Boznik]

Is B=I?

[Walter Roberson]

Is all of B known or just the diagonal elements? Is B all zero except at some diagonal elements?

When you describe A I am not sure whether you are saying that all of the elements of A are known, or if what is known about it is that it is symmetric and singular?

[Suvranil]

All elements of A is known. And only diagonal elements of B are known. As an example, you can consider: [1 -1/2 -1/2; -1/2 1 -1/2; -1/2 -1/2 1] and diag B=[ 3 3 3].

[Bruno Luong]

See my code below that gives

B =

    3.0000    2.0000    2.0000
    2.0000    3.0000    2.0000
    2.0000    2.0000    3.0000

Answer 1

[Bruno Luong]

% Generate testexample of A, B (n x n) matrix such that A*B=A
% Here A is generated to be symmetric but it doesn't matter
n = 5;
C = rand(n,1)*rand(1,n);
K = null(C);
A = K*K.';
B = C.' + eye(n)

% INPUT
dB = diag(B);
clear B

% Reconstruct (off-diagonal elemenst of) B from A and dB
X = A.*(1-dB(:).');
n = size(A,2);
B = diag(dB);
for j=1:n
    i = [1:j-1,j+1:n];
    B(i,j) = A(:,i) \ X(:,j);
end
B