Linear transformations that preserve certain positivity classes of matrices

For each of several S ⊆ R n , n , those linear transformations L : R n,n → R n,n which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M -matrices, the totally nonnegative matrices, the D -stable matrices, the matrices with positive diagonal Lyapunov solutions, and the H -matrices, as well as other related classes. The set of transformations is somewhat different from case to case, but the strategy of proof, while differing in detail, is similar.