Maximum Entropy Elements in the Intersection of an Affine Space and the Cone of Positive Definite Matrices

It is shown that for given positive definite $A$ and $B$ and a linear subspace $\cal{W}$ consisting of $ n \times n$ indefinite (or trivial) Hermitian matrices, there exists a unique positive definite matrix $F$ in $A+\cal{W}$ such that $F^{-1} -B \in \\cal{W}^{\perp}$. This matrix $F$ appears as the maximizer of a certain entropy function. The theorem generalizes a result on Gaussian measures with prescribed margins. Several special cases are presented, yielding new results and recovering known matrix completion results. In case $\cal{W}$ is a coordinate subspace, algorithms to find the optimal $F$ are described and numerical results are presented.