Do you really need to get the inverse of that weight matrix?

Motivation When you run simulations with spatially dependent data, you can use a weight matrix to quantify the nature and degree of dependence between observations. For example, suppose you would like to generate spatially correlated shocks (errors), (\varepsilon), specified as follows: [ \varepsilon = \lambda W \cdot \varepsilon + u ] where (W) is the weight matrix, (\lambda \in [0,1]) is a parameter that determines the degree of spatial dependence, and (u \sim N(0,\sigma^2)).