Nearly positive matrices

Nearly positive matrices are nonnegative matrices which, when premultiplied by orthogonal matrices as close to the identity as one wishes, become positive. In other words, all columns of a nearly positive matrix are mapped simultaneously into the interior of the nonnegative cone by multiplication by a sequence of orthogonal matrices converging to the identity. In this paper, nearly positive matrices are analyzed and characterized in several cases. Some necessary and some sufficient conditions for a nonnegative matrix to be nearly positive are presented. A connection to completely positive matrices is also presented.