How to calculate A matrix if C1=AC2A' ?

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Ahmad Fakih 2019-4-19

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编辑： Ahmad Fakih 2019-4-20

Dear members,

How to calculate matrix A using matlab if C1 and C2 are known?

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David Wilson 2019-4-20

编辑：David Wilson 2019-4-20

在 MATLAB Online 中打开

Any other conditions, such as symmetry, positive definiteness etc?

Have you looked at the Riccati equation and friends?

Below I've tried a crude numerical minimisation strategy, but it is, as expected, very sensitive.

n = 3; % start small, and then work up to say n=10. 
Ax = rand(n,n); % true value 
C2 = rand(n,n); 
C1 = Ax*C2*Ax'; 
f = @(A) norm(A*C2*A' - C1)
g = @(a) f(reshape(a,n,n)); 
optns = optimoptions('fminunc','MaxIterations',1e3,'MaxFunctionEvaluations',1e5)
a0 = randn(n^2,1); % start guess for the elements of [A] in a column. 
a = fminunc(g,a0,optns)
A = reshape(a); 

I've also tried a symbolic version with just 2*2 and random numbers. I get multiple solutions, so are you sure this problem is well formed, say with a unique solution?

n = 2; % test dimension 
syms a11 a12 a21 a22 real 
A = [a11, a12; a21 a22]; 
Ax = [1,2;3,5];   % true solution
C2 = [4,-2;0,6];  % arbitrary constants 
C1 = Ax*C2*Ax'; 
Asoln = solve(A*C2*A'==C1,A) % symbolic toolbox 
for i = 1:length(Asoln.a11) % look at each solution & residual
  Ar = double([Asoln.a11(i), Asoln.a12(i); 
               Asoln.a21(i), Asoln.a22(i)]); 
  res = Ar*C2*Ar' - C1
end 

Ahmad Fakih 2019-4-20

编辑：Ahmad Fakih 2019-4-20

Yeah sorry forgot to tell you the matrices are Symmetrical. @David thanks for the interesting answer. Anyway I found an exact solution, but this requires C2 to be a diagonal ones matrix but that's not a problem for me. Also, cov is basically C1.Thanks all!