How can find the solution of this matrix system?

Hi all,

I'm pretty new in Matlab. I would like to solve this matrix system

XAX' = B

where A is a symmetric positive definite matrix and B is a diagonal matrix with positive elements on the diagonal. How can find X?

Thanks,

Johnny

Answer 1

[Ameer Hamza]

Try fsolve(): https://www.mathworks.com/help/releases/R2020a/optim/ug/fsolve.html

sol = fsolve(@(X) X*A*X.'-B, rand(size(A)))

[Johnny Scalera]

Hi Ameer,

thank you for your answer. Your suggestion is a numerical solution of the previous equation. I was looking for an analytical solution. For example, if A is an identity matrix I, then the system equation turns to be

XAX' = B;
XX' = B;

which can be solved with a Cholesky factorization of B

X = chol(B);

But what appens to the system, if A is a symmetric positive definite matrix and B a diagonal matrix? Such as

A = [a b; b c];
B = [d 0; 0 e];

Thanks,

Johnny

[Ameer Hamza]

For analytical solutions, we usually use the symbolic toolbox. However, as far as I know, the symbolic engine still does not support solving the matrix equation. Even for the following specific cases, there is no solution

syms a b c d e f

A = [a b; b c];
B = [d 0; 0 e];
x = sym('x', size(A));

solve(x*A*x'==B, x)
solve(x*x'==B, x) % A = eye(2)

You may need to try some other symbolic engine (mathematics or maple maybe), but I haven't tried those for matrix equations, so I don't know for sure.