Given a set of $n$ points on which a triangulation is performed, it is possible to construct coefficients $\lambda_{ij}>0$ such that each point $x_i$ is a convex combination of the points connected to it. Suppose $N(i)$ is a set of points adjacent to $x_i$ in a triangulation; then, for each $x_i$,
$$\sum_{j\in N(i)} \lambda_{ij}x_j = x_i $$
and
$$\sum_{j\in N(i)} \lambda_j = 1.$$
In matrix form,
$$\Lambda x=x$$
where $\Lambda$ is a square, $n\times n$, row stochastic matrix (non-negative entries, and no self-loops).
Now suppose matrix $\Lambda$ is given. Can one reach configuration $x$ by the Power Method, i.e., repeatedly multiplying $\Lambda$ by a vector? I am asking because the above matrix form reminds me of the eigenvalue/eigenvector definition. If not, what would be a good approach to compute such a configuration? Is the solution unique?
One more interesting question: can the coefficients $\lambda_{ij}$ be constructed in such a way that $\lambda_{ij} = \lambda_{ji}$, i.e., the matrix $\Lambda$ is symmetric, that is to say doubly stochastic? This way of constructing coefficients means that the Power Method should converge to $x$.