Maximum Entropy Elements in the Intersection of an Affine Space and the Cone of Positive Definite Matrices
Mihály Bakonyi, Hugo J. Woerdeman · SIAM Journal on Matrix Analysis and Applications · 1995
It is shown that for given positive definite A and B and a linear subspace $\mathcal{W}$ consisting of $n \times n$ indefinite (or trivial) Hermitian matrices, there exists a unique positive definite matrix F in $A + \mathcal{W}$ such that $F^{ - 1} - B \in \mathcal{W}^ \bot $. This matrix F appears as the maximizes 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 $\mathcal{W}$ is a coordinate subspace, algorithms to find the optimal F are described and numerical results are presented.