Gersgorin Circle Theorem

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A Gersgorin Circle Theorem is a complex plane region that contains all the eigenvalues of a complex square matrix.



References

2021

$R_i=\displaystyle \sum_{j=1, i \ne j}^n \left|a_{ij}\right|$

(1)
Then each eigenvalue of $\mathbf{A}$ is in at least one of the disks

$\{z:\left|z-a_{ii}\right|\leq R_i\}$

(2)
The theorem can be made stronger as follows. Let $r$ be an integer with $1\leq r \leq n$, and let $S_j^{(r-1)}$ be the sum of the magnitudes of the $r-1$ largest off-diagonal elements in column $j$. Then each eigenvalue of $\mathbf{A}$ is either in one of the disks

$\{z:\left|z-a_{jj}\right|\leq S_j^{(r-1)}\}$,

(3)
or in one of the regions

$\{z:\displaystyle\sum_{i\in P} \left|z-a_{ii}\right| \leq \displaystyle\sum_{i \in P}R_i\}$

(4)
where $P$ is any subset of $\{1,2,\cdots,n\}$ such that $\left|P\right|=r$ (Brualdi and Mellendorf 1994).

2019

1994

  • (Brualdi & Mellendorf, 1994)R.A. Brualdi, and S. Mellendorf (1994)."Regions in the Complex Plane Containing the Eigenvalues of a Matrix." Amer. Math. Monthly 101, 975-985.