  For discrete random variables X_1,..., X_n we construct an n by n matrix. In
the (i,j) entry we put the mutual information I(X_i;X_j) between X_i and X_j.
In particular, in the (i,i) entry we put the entropy H(X_i)=I(X_i;X_i) of X_i.
This matrix, called the mutual information matrix of (X_1,...,X_n), has been
conjectured to be positive semi-definite. In this note, we give counterexamples
to the conjecture, and show that the conjecture holds for up to three random
variables.
