## Geršgorin and His CirclesTheGer? sgorin CircleTheorem, averywell-known resultin linear algebra today, stems from the paper of S. Ger? sgorin in 1931 (which is reproduced in AppendixD)where,givenanarbitraryn×ncomplexmatrix,easyarithmetic operationsontheentriesofthematrixproducendisks,inthecomplexplane, whose union contains all eigenvalues of the given matrix. The beauty and simplicity of Ger? sgorin’s Theorem has undoubtedly inspired further research in this area, resulting in hundreds of papers in which the name “Ger? sgorin” appears. The goal of this book is to give a careful and up-to-date treatment of various aspects of this topic. The author ?rst learned of Ger? sgorin’s results from friendly conversations with Olga Taussky-Todd and John Todd, which inspired me to work in this area.Olgawasclearlypassionateaboutlinearalgebraandmatrixtheory,and her path-?nding results in these areas were like a magnet to many, including this author! It is the author’s hope that the results, presented here on topics related to Ger? sgorin’s Theorem, will be of interest to many. This book is a?ectionately dedicated to my mentors, Olga Taussky-Todd and John Todd. There are two main recurring themes which the reader will see in this book. The ?rst recurring theme is that a nonsingularity theorem for a mat- ces gives rise to an equivalent eigenvalue inclusion set in the complex plane for matrices, and conversely. Though common knowledge today, this was not widely recognized until many years after Ger? sgorin’s paper appeared. That these two items, nonsingularity theorems and eigenvalue inclusion sets, go hand-in-hand, will be often seen in this book. |

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### Contents

1 Basic Theory | 1 |

2 GerˇsgorinType Eigenvalue InclusionTheorems | 35 |

3 More Eigenvalue Inclusion Results | 73 |

4 Minimal Gerˇsgorin Sets and TheirSharpness | 97 |

5 GFunctions | 127 |

6 GerˇsgorinType Theorems for PartitionedMatrices | 155 |

Appendix A Gerˇsgorins Paper from 1931and Comments on His Life and Research | 189 |

Appendix B Vector Norms and InducedOperator Norms | 199 |

Appendix C The PerronFrobenius Theory ofNonnegative Matrices MMatrices andHMatrices | 201 |

Appendix D Matlab 6 Programs | 205 |

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### Common terms and phrases

A G Cnxn A I aid ai,i ai,j Appendix assume Brauer Cassini ovals Brauer set Cn×n compact set complex number complex plane convex set deﬁned Deﬁnition Deﬁnition 5.1 denote diagonal entries diagonal matrix diagonally dominant matrix directed graph disjoint eigenvalue inclusion results eigenvalue inclusion set equivalent extended complex plane f G Qn ﬁnal ﬁrst ﬁxed follows G-functions Gersgorin disks Given a partition gives Hence i,j G N implies irreducible matrix lC(A Lemma lf(A minimal Gersgorin set minimal point nonnegative matrix nonsingular M-matrix nonsingularity result nonzero norm normal reduced form notation off-diagonal entries operator norm Ostrowski partition Tr partitioned by Tr partitioned matrix permutation matrix proof of Theorem Q5 G real numbers remark result of Theorem ri(A Rnxn satisﬁes shown in Fig singular M-matrix strictly diagonally dominant strong cycle subset TLXTL triangle inequality valid Varga vector vertex weak cycles X,Y]=meshgrid(x,y Znxn