Solutions of Z-matrix equations.

*(English)*Zbl 0652.15016The solvability of the matrix equation \(Ax=b\), where A is a Z-matrix and b is a nonnegative vector is discussed. A previous existence theorem for a solution is reproved. The results are applied to study nonnegative vectors in the range of Z-matrices. It is a characteristic of the problem described above that the existence and the nature of a solution depend entirely on graph theoretic conditions. In the case that nonnegativity of the solution is not required, it is shown that there are no purely graph theoretic conditions for solvability but, however, there are graph theoretic results concerning the nature of the solution.

Reviewer: M.de la Sen

##### MSC:

15B48 | Positive matrices and their generalizations; cones of matrices |

15A06 | Linear equations (linear algebraic aspects) |

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\textit{D. Hershkowitz} and \textit{H. Schneider}, Linear Algebra Appl. 106, 25--38 (1988; Zbl 0652.15016)

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##### References:

[1] | Carlson, D., A. note on M-matrix equations, SIAM j., 11, 213-217, (1963) |

[2] | D. Hershkowitz, U.G. Rothblum, and H. Schneider, Characterizations and classifications of M-matrices using generalized nullspace, Linear Algebra Appl., to appear. · Zbl 0654.15015 |

[3] | Hershkowitz, D.; Schneider, H., On the generalized nullspace of M-matrices and Z-matrices, Linear algebra appl., 106, 5-23, (1988) · Zbl 0647.15005 |

[4] | Schneider, H., The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey, Linear algebra appl., 84, (1986) · Zbl 0613.15017 |

[5] | Victory, H.D., On nonnegative solutions to matrix equations, SIAM J. algebraic discrete methods, 6, 406-412, (1985) · Zbl 0586.15003 |

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