There is a one to one mapping between a p dimensional strictly positive definite covariance
matrix Σ and its matrix logarithm L. We exploit this relationship to study the
structure induced on Σ through a sparsity constraint on L. Consider L as a random
matrix generated through a basis expansion, with the support of the basis coefficients
taken as a simple random sample of size s = s
∗
from the index set [p(p + 1)/2] =
{1, . . . , p(p + 1)/2}. We find that the expected number of non-unit eigenvalues of Σ, denoted
E[|A|], is approximated with near perfect accuracy by the solution of the equation
4p + p(p − 1)
2(p + 1)
h
log p
p − d

−
d
2p(p − d)
i
− s
∗ = 0.
Furthermore, the corresponding eigenvectors are shown to possess only p − |Ac
| nonzero
entries. We use this result to elucidate the precise structure induced on Σ and Σ−1
.
We demonstrate that a positive definite symmetric matrix whose matrix logarithm is
sparse is significantly less sparse in the original domain. This finding has important
implications in high dimensional statistics where it is important to exploit structure in
order to construct consistent estimators in non-trivial norms. An estimator exploiting
the structure of the proposed class is presented.