Hochstenbach, Michiel E.; Singer, David A.; Zachlin, Paul F. Eigenvalue inclusion regions from inverses of shifted matrices. (English) Zbl 1176.15022 Linear Algebra Appl. 429, No. 10, 2481-2496 (2008). This well-written and rich-in-content paper deals with a classic topic of eigenvalue inclusion problems in a contemporary setting with computational examples. It examines theoretical properties of inclusion sets and their relationships to the harmonic Rayleigh-Ritz technique. In particular, it deals with eigenvalue inclusion regions derived from the field of values, pseudospectra, Gershgorin regions and Brauer regions, respectively, for the inverse of shifted regions of non-singular square matrices. The main result of the paper is that by varying the shift the authors get a family of inclusion regions with surprising properties which are that the intersection of the family is exactly the spectrum, and an appropriate limit of the set converges to the “mother set” of the field of values.Section 1 gives the theoretical setting of a nonsingular complex matrix, its spectrum and its field of values and gives an overview of the paper. Then a modified harmonic Rayleigh-Ritz method is used and the authors observe that the standard Rayleigh-Ritz method can be viewed as the harmonic Rayleigh-Ritz method with target at infinity (Section 2). Section 3 deals with eigenvalue inclusion regions form the field of values of inverses of shifted matrices and derives the main result mentioned above – as Theorem 5 (p. 2484) with geometric interpretation. The authors are commanded for illustrating the Theorem by giving graphs of spectra for the 300 by 300 randcolu test matrix selected from the MATLAB gallery for various shifts. Equivalent theorems are derived for the Gershgorin region (Theorem 8, p. 2488) and Brauer region (Theorem 10, p. 2489), respectively in Section 4. Theorem 13, p. 2490 is the equivalent to Theorem 5, now for pseudospectra. Sections 6 and 7 show that the obtained results yield practical methods to approximate inclusion regions for large matrices and their inverses via subspace approximation techniques. Moreover, Section 7 gives a short high-level abstract algorithm to determine an approximate inclusion region based on fields of values in a Krylov space setting. Further graphs of spectra for the 1000 by 1000 grcar matrix of MATLAB are given.This paper is related to the third author’s Ph.D. thesis [On the field of values of the inverse of a matrix, Cleveland, Ohio: Case Western Reserve University (2007), online at http://www.ohiolink.edu/etd/view.cgi?case1181231690]. A further important reference given in the paper is the book by R. S. Varga [Geršgorin and his circles. Springer Series in Computational Mathematics 36. Berlin: Springer (2004; Zbl 1057.15023)]. Reviewer: Herbert Kreyszig (New York) Cited in 1 Document MSC: 15A42 Inequalities involving eigenvalues and eigenvectors Keywords:inclusion regions; exclusion regions; inclusion curves; exclusion curves; field of values; numerical range; large sparse matrix; Gershgorin regions; ovals of Cassini; Brauer regions; pseudospectra; subspace methods; inverses of shifted matrices; eigenvalue inclusion; harmonic Rayleigh-Ritz method Citations:Zbl 1057.15023 Software:Eigtool; Matlab PDFBibTeX XMLCite \textit{M. E. Hochstenbach} et al., Linear Algebra Appl. 429, No. 10, 2481--2496 (2008; Zbl 1176.15022) Full Text: DOI References: [1] Beattie, C.; Ipsen, I. C.F., Inclusion regions for matrix eigenvalues, Linear Algebra Appl., 358, 281-291 (2003) · Zbl 1054.15020 [2] Gau, H.-L.; Wu, P. Y., Numerical range and Poncelet property, Taiwanese J. Math., 7, 173-193 (2003) · Zbl 1051.15019 [3] Gustafsson, B.; Shapiro, H. S., What is a quadrature domain?, (Quadrature domains and their applications. 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