ALPS: Matrices with nonpositive off-diagonal entries

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices having nonpositive off-diagonal entries. The major classes considered are the sets of diagonally stable, stable, and semipositive matrices, denoted respectively by $\text{A}$, $\text{L}$, and $\text{S}$. The conditions defining these classes are weakened, and the resulting classes are examined. Their relationship to the classes of real matrices $\text{P}$ and $\text{P}$₀, whose off-diagonal entries are nonpositive and whose principal minors are respectively all positive and all nonnegative, is also included.