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| {{dablink|For the [[chemistry]] related meaning of this term see [[Z-matrix (chemistry)]].}}
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| In [[mathematics]], the class of ''Z''-matrices are those [[matrix (mathematics)|matrices]] whose off-diagonal entries are less than or equal to zero; that is, a ''Z''-matrix ''Z'' satisfies
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| :<math>Z=(z_{ij});\quad z_{ij}\leq 0, \quad i\neq j.</math>
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| 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.
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| The [[Jacobian matrix and determinant|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.
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| Related classes are [[L-matrix|''L''-matrices]], [[M-matrix|''M''-matrices]], [[P-matrix | ''P''-matrices]], [[Hurwitz matrix | ''Hurwitz'' matrices]] and [[Metzler matrix |''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-matrix|''P''-matrices]] are nonsingular ''M''-matrices.
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| ==See also==
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| * [[P-matrix]]
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| * [[M-matrix]]
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| * [[Hurwitz matrix]]
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| * [[Metzler matrix]]
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| == References ==
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| * {{cite journal| doi= 10.1016/j.amc.2005.07.050| author=Huan T., Cheng G., Cheng X.| title= Modified SOR-type iterative method for Z-matrices|journal = Applied Mathematics and Computation| volume =175 |issue =1|date= 1 April 2006| pages= 258–268}}
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| * {{cite book | url=http://www-users.cs.umn.edu/~saad/books.html|author= Saad, Y. |title=Iterative methods for sparse linear systems| publisher= Society for Industrial and Applied Mathematics|location= Philadelphia, PA.|edition =2nd | page =28 | isbn=0-534-94776-X}}
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| [[Category:Matrices]]
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| {{Linear-algebra-stub}}
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Latest revision as of 14:49, 8 June 2014
Hello, my name is Andrew and my wife doesn't like it at all. For a while I've been in Mississippi but now I'm considering other choices. He functions as a bookkeeper. To climb is some thing I truly appreciate performing.
my web blog are psychics real (cpacs.org)