Z-matrix (mathematics)
Encyclopedia
In mathematics
, 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 satisfies
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 satisfies
Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix
Quasipositive matrix
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative It is named after the American economist Lloyd Metzler....
, 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
M-matrix
In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are positive. M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices In mathematics, especially linear algebra, an M-matrix is a Z-matrix with...
, P-matrices, Hurwitz matrices
Hurwitz matrix
-Hurwitz matrix and the Hurwitz stability criterion:In mathematics, Hurwitz matrix is a structured real square matrix constructed with coefficientsof a real polynomial...
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.