Volume 12, pp. 113-133, 2001.

Geršgorin-type eigenvalue inclusion theorems and their sharpness

Richard S. Varga

Abstract

Here, we investigate the relationships between ${\cal G} (A)$, the union of Geršgorin disks, ${\cal K} (A)$, the union of Brauer ovals of Cassini, and ${\cal B} (A)$, the union of Brualdi lemniscate sets, for eigenvalue inclusions of an $n \times n$ complex matrix $A$. If $\sigma (A)$ denotes the spectrum of $A$, we show here that \[ \sigma (A) \subseteq {\cal B} (A) \subseteq {\cal K} (A) \subseteq {\cal G} (A) \] is valid for any weakly irreducible $n \times n$ complex matrix $A$ with $n \geq 2$. Further, it is evident that ${\cal B} (A)$ can contain the spectra of related $n \times n $ matrices. We show here that the spectra of these related matrices can fill out ${\cal B} (A)$. Finally, if ${\cal G}^{\cal R} (A)$ denotes the minimal Geršgorin set for $A$, we show that \[ {\cal G}^{\cal R} (A) \subseteq {\cal B} (A). \]

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Key words

Geršgorin disks, Brauer ovals of Cassini, Brualdi lemniscate sets, minimal Geršgorin sets.

AMS subject classifications

15A18.

Links to the cited ETNA articles

[7]Vol. 8 (1999), pp. 15-20 Richard S. Varga and Alan Krautstengl: On Geršgorin-type problems and ovals of Cassini

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