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In mathematics, the class of *Z*-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form:

Note 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.

In the context of quantum complexity theory, these are referred to as *stoquastic operators*.^{[1]}

- Huan T.; Cheng G.; Cheng X. (1 April 2006). "Modified SOR-type iterative method for Z-matrices".
*Applied Mathematics and Computation*.**175**(1): 258–268. doi:10.1016/j.amc.2005.07.050. - Saad, Y. (1996).
*Iterative methods for sparse linear systems*(2nd ed.). Philadelphia, PA.: Society for Industrial and Applied Mathematics. p. 28. ISBN 0-534-94776-X. - Berman, Abraham; Plemmons, Robert J. (2014).
*Nonnegative Matrices in the Mathematical Sciences*. Academic Press. ISBN 9781483260860.