Inverses of Perron complements of inverse \(M\)-matrices.

*(English)*Zbl 0959.15023The concept of the Perron complement of a nonnegative and irreducible matrix was introduced by C. D. Meyer [Linear Algebra Appl. 114/115, 69-94 (1989; Zbl 0673.15006) and SIAM Rev. 31, No. 2, 240-272 (1989; Zbl 0685.65129)]. In the present paper the author studies the properties of the Perron complement of a matrix which is an inverse of an irreducible \(M\)-matrix. He shows that such complement is again an inverse \(M\)-matrix. He also investigates the directed graph of the inverses of the extended Perron complements (associated to an inverse of an irreducible \(M\)-matrix), and shows that the Perron complement of an inverse of an irreducible tridiagonal \(M\)-matrix is again an inverse of an irreducible tridiagonal \(M\)-matrix and hence totally nonnegative.

Reviewer: Philippe Elbaz-Vincent (Montpellier)

##### MSC:

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

15A09 | Theory of matrix inversion and generalized inverses |

##### Keywords:

nonnegative matrix; inverse \(M\)-matrix; Schur and Perron complements; irreducible matrix; directed graph of a matrix
PDF
BibTeX
XML
Cite

\textit{M. Neumann}, Linear Algebra Appl. 313, No. 1--3, 163--171 (2000; Zbl 0959.15023)

Full Text:
DOI

##### References:

[1] | A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994 · Zbl 0815.15016 |

[2] | A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1966 |

[3] | Crabtree, D.E., Applications of M-matrices to nonnegative matrices, Duke math. J., 33, 197-208, (1966) · Zbl 0142.27101 |

[4] | R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985 · Zbl 0576.15001 |

[5] | G. Golub, C. Van Loan, Matrix Computation, third ed., The John Hopkins University Press, Baltimore, MD, 1996 · Zbl 0865.65009 |

[6] | C.R. Johnson, C. Xenophotos, Irreducibility and primitivity of Perron complements: applications of the compressed graph, in: Graph Theory and Sparse Matrix Computations, IMA Vol. Math. Appl., vol. 56, 1993, pp. 101-106 |

[7] | M. Lewin, Totally nonnegative, M-, and Jacobi matrices, SIAM J. Algebraic Discrete Methods 4 (1980) 419-421 |

[8] | Lewin, M.; Neumann, M., On the inverse M-matrix problem for \((0,1)\)-matrices, Linear algebra appl., 30, 41-50, (1980) · Zbl 0434.05051 |

[9] | Meyer, C.D., Uncoupling the Perron eigenvector problem, Linear algebra appl., 114/115, 69-94, (1989) · Zbl 0673.15006 |

[10] | Meyer, C.D., Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM rev., 31, 240-272, (1989) · Zbl 0685.65129 |

[11] | Schneider, H., Theorems on M-splittings of a singular M-matrix which depend on graph structure, Linear algebra appl., 58, 407-424, (1984) · Zbl 0561.65020 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.