  Usually, opinion formation models assume that individuals have an opinion
about a given topic which can change due to interactions with others. However,
individuals can have different opinions in different topics and therefore
n-dimensional models are best suited to deal with these cases. While there have
been many efforts to develop analytical models for one dimensional opinion
models, less attention has been paid to multidimensional ones. In this work, we
develop an analytical approach for multidimensional models of continuous
opinions where dimensions can be correlated or uncorrelated. We show that for
any generic reciprocal interactions between agents, the mean value of initial
opinion distribution is conserved. Moreover, for positive social influence
interaction mechanisms, the variance of opinion distributions decreases with
time and the system converges to a delta distributed function. In particular,
we calculate the convergence time when agents get closer in a discrete quantity
after interacting, showing a clear difference between correlated and
uncorrelated cases.
