No, the answer is not unique. There are many ways to show this. One possibility is to notice that spectral decomposition of a square $p$ by $p$ matrix $X$ is the solution to the maximization of a convex function of $w$. Consider the first eigen-vector/value:
(where $\lambda_1$ is the first eigen-value and $w^*$ the first eigen-vector).
The solution to such problems (e.g. the values of $w$ attaining that maximum) are, in general, not unique.
However the algorithms for computing these solutions are deterministic, meaning that save for numerical corner cases, the solutions you get should be the same.
Example of such numerical corner cases: cases where several eigen-values are (numerically) the same, cases where the $X$ is rank-deficient...