the class of

*Z*-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, a*Z*-matrix*Z*satisfiesNote that this definition coincides precisely with that of a

**negated**Metzler matrix or quasipositive matrix, thus the term*quasinegative*matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made.The Jacobian of a

**competitive**dynamical system is a*Z*-matrix by definition. Likewise, if the Jacobian of a**cooperative**dynamical system is*J*, then (−*J*) is a*Z*-matrix.Related classes are

*L*-matrices,*M*-matrices,*P*-matrices,*Hurwitz*matrices and*Metzler*matrices.*L*-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a*Z*-matrix is an*M*-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both*Z*-matrices and*P*-matrices are nonsingular*M*-matrices.