I have an equation of the form $J^Te=f$, where $e$ and $f$ are known vectors and $J$ is an unknown matrix. How can I efficiently compute $J^T(JJ^T)^{-1/2}e$ ?My motivation to address this problem is that, as a part of a bigger problem I need to compute matrix $J$ then compute $J^T(JJ^T)^{-1/2}e$. However, computation of matrix $J$ is expensive and $J$ results to be a large matrix, thereby the determination of product-transpose-squareroot-inverse of it is taking a lot of time. Hence, I'm planning to bypass the expensive computation of $J$ with the help some known vectors $e$ and $f$ and the relation $J^Te=f$.
Fast computation of square root inverse of matrix, matrix being determined from Ax=b form
linear algebra;matrix;inverse;matrix free