Question: Cholesky Decomposition for a symbolic matrix

I wrote down a covariance matrix through the following command:

 A := Matrix([[a, b], [b, c]], shape = symmetric, attributes = [positive_definite]);

and then I computed the Cholesky decomposition of this matrix using afterwards the command simplify as well

B := LUDecomposition(A, method = Cholesky)

C:=simplify(%)

What I obtained is this matrix

Matrix(2, 2, {(1, 1) = sqrt(a), (1, 2) = 0, (2, 1) = conjugate(b)/sqrt(a), (2, 2) = sqrt(c-abs(b^2/a))})

Is there any way in which I can tell the software that some of the entries are positive and real, so that I don't obtain any absolute value or conjugate??

The problem is that when I then multiply C with its transpose I don't exactly obtain the matrix A, as it should be.. It is the same but it is written in terms of conjugates and absolute values.

Can you help me?
Manuele