## Abstract

For nonnegative matrices A, the well known Perron-Frobenius theory studies the spectral radius ρ(A). Rump has offered a way to generalize the theory to arbitrary complex matrices. He replaced the usual eigenvalue problem with the equation |Ax| = λ|x| and he replaced ρ(A) by the signed spectral radius, which is the maximum λ that admits a nontrivial solution to that equation. We generalize this notion by replacing the linear transformation A by a map f:Cn→R whose coordinates are seminorms, and we use the same definition of Rump for the signed spectral radius. Many of the features of the Perron-Frobenius theory remain true in this setting. At the center of our discussion there is an alternative theorem relating the inequalities f(x) ≥ λ|x| and f(x) < λ|x|, which follows from topological principals. This enables us to free the theory from matrix theoretic considerations and discuss it in the generality of seminorms. Some consequences for P-matrices and D-stable matrices are discussed.

Original language | English |
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Pages (from-to) | 245-284 |

Number of pages | 40 |

Journal | Linear Algebra and Its Applications |

Volume | 399 |

Issue number | 1-3 |

DOIs | |

State | Published - 1 Apr 2005 |

Event | International Meeting on Matrix Analysis and Applications - Ft. Lauderdale, Fl, United States Duration: 14 Dec 2003 → 16 Dec 2003 |

### Funding

Funders | Funder number |
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National Science Foundation | |

Directorate for Mathematical and Physical Sciences | 0201333 |

## Keywords

- Nonnegative matrices
- P-matrices
- Seminorms
- Theorems on alternative