  In this paper we establish links between, and new results for, three problems
that are not usually considered together. The first is a matrix decomposition
problem that arises in areas such as statistical modeling and signal
processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix
and an unknown low rank positive semidefinite matrix, decompose $X$ into these
constituents. The second problem we consider is to determine the facial
structure of the set of correlation matrices, a convex set also known as the
elliptope. This convex body, and particularly its facial structure, plays a
role in applications from combinatorial optimization to mathematical finance.
The third problem is a basic geometric question: given points
$v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered
ellipsoid passing \emph{exactly} through all of the points.
  We show that in a precise sense these three problems are equivalent.
Furthermore we establish a simple sufficient condition on a subspace $U$ that
ensures any positive semidefinite matrix $L$ with column space $U$ can be
recovered from $D+L$ for any diagonal matrix $D$ using a convex
optimization-based heuristic known as minimum trace factor analysis. This
result leads to a new understanding of the structure of rank-deficient
correlation matrices and a simple condition on a set of points that ensures
there is a centered ellipsoid passing through them.
