I have square matrices $X$,$A$ and $X'X-A=0$. $A$ is given and is positive definite and I need to get matrix $X$. I know $X$ is not unique since $TX$ such that $T'T=I$ will satisfy. My problem is that is it possible to combine the equation with some prior knowledge (some sort of restriction) such as symmetry, triangularity, sparsity and somehow get a unique $X$.
Many thanks!
A positive definite matrix $A$ has a unique positive definite square root. If we diagonalize $A$ as $A = U D U^*$ with $U$ a unitary matrix and $D$ diagonal, then this square root is $X = U \sqrt{D} U^*$ where $\sqrt{D}$ is the diagonal matrix whose elements are the square roots of the corresponding elements of $D$.
If $A$ is real, $U$ is orthogonal and $X$ is real and symmetric.