Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis

One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rankr, has positive (Lebesgue) measure if and only ifr is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.