### Abstract

Bergstrom's inequality is generalized. Using the new inequality, several interesting results concerning Hadamard product of matrices are proved. More specifically, let P,Q ε{lunate} C^{N×N} such that P>0 and Q≥0, Q_{ii}>0. Then we prove the following tight inequalities: (a) Q_{ii}[(P⊙Q)^{-1}]_{ii}≤ (P^{-1}_{ii}; (b) Q_{ii}{[Re(P⊙Q)]^{-1}}_{ii} ≤ max {[Re(φPφ^{*})]^{-1}}_{ii}; (c) det(P⊙Q)≥ (detP)π^{N}_{i=1} Q_{ii}; (d) det Re(P⊙ Q) ≥ min det[Re(φPφ^{*})]π^{N}_{i=1} Q_{ii}, where in (b) and (d) maximization and minimization are over all unitary diagonal matrices φε{lunate}C^{N×N}.

Original language | English (US) |
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Pages (from-to) | 135-151 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 161 |

Issue number | C |

DOIs | |

State | Published - Jan 15 1992 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

Fong Yau, S., & Bresler, Y. (1992). A generalization of bergstrom's inequality and some applications.

*Linear Algebra and Its Applications*,*161*(C), 135-151. https://doi.org/10.1016/0024-3795(92)90009-Y