Z-matrix (mathematics)

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For the chemistry related meaning of this term see Z-matrix (chemistry).

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

Z=(z_{ij});\quad z_{ij}\leq 0, \quad i\neq j.

Note that this definition coincides precisely with that of a negated 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.

Two related classes are L-matrices and M-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.

[edit] References

  • Huan T., Cheng G., Cheng X., Modified SOR-type iterative method for Z-matrices. Applied Mathematics and Computation, Volume 175 Issue 1, 1 April 2006, pages 258-268.
  • Saad, Y. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics. Philadelphia, PA. 2nd edition. page 28.


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