The Geršgorin Circle Theorem, a fundamental result in linear algebra, originated from S. Geršgorin's 1931 paper (included in Appendix D). This theorem demonstrates that simple arithmetic operations on the entries of any n×n complex matrix yield disks in the complex plane, collectively encompassing all eigenvalues of the matrix. Its elegance has spurred extensive research, leading to numerous papers featuring Geršgorin's name. This book aims to provide a thorough and contemporary exploration of various facets related to this theorem. The author was first introduced to Geršgorin's work through engaging discussions with Olga Taussky-Todd and John Todd, whose enthusiasm for linear algebra and matrix theory greatly influenced the author's own research journey. The author hopes that the topics presented will resonate with many readers. The book is dedicated to the author's mentors, Olga Taussky-Todd and John Todd. Two main themes recur throughout the text: the relationship between nonsingularity theorems for matrices and their corresponding eigenvalue inclusion sets in the complex plane. While now widely acknowledged, this connection was not fully appreciated until long after Geršgorin's original findings, and the interplay between these concepts is a central focus of the book.
