I have a sampling problem in which I need the number of antecedent of a symmetric definite positive matrix A by a function of the shape $f \colon B \mapsto BB^T$. All matrices are square in my application.
Reformulating the question, I wish to know for a fixed square matrix $A$ how many (if it's a finite number) square matrix $B$ satisfying $BB^T = A$ are there?
I know the decomposition is not unique since the Cholesky and the singular value decomposition generally yield two different $B_1$, $B_2$ satisfying the above condition. I am having trouble checking previous work on the subject since this decomposition doesn't seem to have a name! If anyone knows a way to think about this question, or has a relevant source to flag, I'm all ears.
If $U$ is any orthogonal matrix, namely if $UU^T=I$, then $BU(BU)^T=BUU^TB^T=BB^T$.
Therefore, without other restrictions on $B$ you will have infinitely many, in dimension larger than $1$.