according to answerd questions: https://math.stackexchange.com/a/2858336/305862 and How to find a matrix that produced a symmetric matrix?
to find a matrix X which produces a given symmetric matrix A at equation:
$$ X^{T} X = A $$
we can decompose A as:
$$ A = O DO^{T} = O \sqrt{D} \sqrt{D} O^{T} = (\sqrt{D} {O}^T)^{T} \sqrt{D} O^{T} $$
so : $$ X = \sqrt{D} {O}^T $$ is the answer where D is diagonal eigen value matrix and O is the orthogonal eigen vector.
for proof I made a supposed matrix as follow:
$$ X=\begin{bmatrix} 1.2 & -0.1 & 0.2\\ 0.2 & 0.9 & 0.1 \\ -0.1 & -0.3 & 1.1 \end{bmatrix} $$
so the A matrix will be :
$$ A=\begin{bmatrix} 1.49 & 0.09 & 0.15 \\ 0.09 & 0.91 & -0.26 \\ 0.15 & -0.26 & 1.26 \end{bmatrix} $$
accordingly the D and O matrices are as follow:
$$ D=\begin{bmatrix} 1.56657 & 0 & 0 \\ 0 & 0.741073 & 0 \\ 0 & 0 & 1.35236 \end{bmatrix} $$
$$ O=\begin{bmatrix} -0.869483 & 0.199387 & -0.451934 \\ 0.0742032 & -0.851811 & -0.518567 \\ -0.488358 & -0.48442 & 0.72584 \end{bmatrix} $$
for proof the result of : $$ O DO^{T} $$ is exactly as the A matrix.
but the result of: $$ \sqrt{D} {O}^T = \begin{bmatrix} -1.08827 & 0.0928747 & -0.611241 \\ 0.171643 & -0.733286 & -0.417016 \\ -0.525558 & -0.603047 & 0.844086 \end{bmatrix} $$
that is different from X matrix!
I would be thankful for any idea that can explain why so is and how to correct the answer.