Geršgorin-type eigenvalue inclusion theorems and their sharpness

Varga, Richard S.

Electronic transactions on numerical analysis, Tome 12 (2001), pp. 113-133

Résumé

Here, we investigate the relationships between $G(A)$, the union of Ger$\check $sgorin disks, $K(A)$, the union of Brauer ovals of Cassini, and $B(A)$, 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 $\sigma (A) \subseteq B(A) \subseteq K(A) \subseteq G(A)$ is valid for any weakly irreducible n $\times n$ complex matrix A with n $\geq 2$. Further, it is evident that $B(A)$ can contain the spectra of related n $\times n$ matrices. We show here that the spectra of these related matrices can fill out $B(A)$.

Détail

Classification : 15A18
Keywords: ger$\check $sgorin disks, Brauer ovals of Cassini, brualdi lemniscate sets, minimal ger$\check $sgorin sets