How to solve matrix equation for variable inside a summation? Analytical or numerical.

Hello together

is there a chance to solve the following matrix equation for X?

where all Ai and D are known. X is of course constant over iteration, but unknown. All are 6x6 matrices.

I would be happy on any hint of any analytical, numerical, or what so ever solution.

Thank you!

Answer 1

[infinity]

Hello,

There are several ways that you can apply. Here, I just show you an example how to solve this problem by using solve function in matlab

clear
A = [1 2; 3 4]
syms x1 x2 x3 x4
X = [x1 x2; x3 x4]
B = A*X*A'
D = [1 0; -2 3]

sol = solve(B==D)

sol.x1
sol.x2
sol.x3
sol.x4

In this case, I assume that "n = 1", A, X, D are 4x4 matrices, but you can extend for 6x6 or larger and different n.

Hope it could help you.

Answer 2

[Pablo Noever]

Hey,

thanks for the fast answer. Worked out for me!

I have transfered that to my problem just in case anyone looks for a similar solution:

n = 10;

A = rand(6,6,n);

% Reference Result
X_ref = rand(6);
RES_ref=zeros(6);
for i = 1:n
    RES_ref = RES_ref + A(:,:,i) * X_ref * A(:,:,i)';
end

% Calculation
X_sym = sym('x%d%d', [6 6]);
RES_sym = zeros(6);

for i = 1:n
    RES_sym = RES_sym + A(:,:,i) * X_sym * A(:,:,i)';
end

    % Solution
    S = solve(RES_sym == RES_ref);
   
% Result
    X_sol = zeros(6);
    for j = 1:6
        for k = 1:6
            X_sol(j,k) = double(S.(sprintf('x%d%d',j,k)));
        end
    end
    
% Should give zero
Check = sum(sum(X_sol - X_ref));

Cheers!

Pablo