  The question of when zeros (i.e., sparsity) in a positive definite matrix $A$
are preserved in its Cholesky decomposition, and vice versa, was addressed by
Paulsen et al. in the Journal of Functional Analysis (85, pp151-178). In
particular, they prove that for the pattern of zeros in $A$ to be retained in
the Cholesky decomposition of $A$, the pattern of zeros in $A$ has to
necessarily correspond to a chordal (or decomposable) graph associated with a
specific type of vertex ordering. This result therefore yields a
characterization of chordal graphs in terms of sparse positive definite
matrices. It has also proved to be extremely useful in probabilistic and
statistical analysis of Markov random fields where zeros in positive definite
correlation matrices are intimately related to the notion of stochastic
independence. Now, consider a positive definite matrix $A$ and its Cholesky
decomposition given by $A = LDL^T$, where $L$ is lower triangular with unit
diagonal entries, and $D$ a diagonal matrix with positive entries. In this
paper, we prove that a necessary and sufficient condition for zeros (i.e.,
sparsity) in a positive definite matrix $A$ to be preserved in its associated
Cholesky matrix $L$, \, and in addition also preserved in the inverse of the
Cholesky matrix $L^{-1}$, is that the pattern of zeros corresponds to a
co-chordal or homogeneous graph associated with a specific type of vertex
ordering. We proceed to provide a second characterization of this class of
graphs in terms of determinants of submatrices that correspond to cliques in
the graph. These results add to the growing body of literature in the field of
sparse matrix decompositions, and also prove to be critical ingredients in the
probabilistic analysis of an important class of Markov random fields.
