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that is the problem with 2 norm, that it entails computing $A^{1/2}z$ which yet remains the problem. That is why in computation of $A^{-1}b$ people use $A$ norm (e.g. conjugate gradient method) cause it overcomes the problem of computing $A^{-1}z$ where z remains an approximation of x
As Igor pointed , there can be a lot of $S$ possible , which also say that there can be a number of $y$ and $S$ exist for pair of $A$ and $z$. What I'm trying to do is, suppose by some contraption I generated a $y$, (without explicitly finding out $S$ ) , I want to verify that for such $y$, there indeed exist some $S$ satisfying $A=SS^{T}$ ( again I'm not interested in calculation of S,existence is sufficient )
