Gersgorin Circle Theorem
A Gersgorin Circle Theorem is a complex plane region that contains all the eigenvalues of a complex square matrix.
- AKA: Gershgorin Circle Theorem.
- …
- Example(s):
- https://en.wikipedia.org/wiki/Gershgorin_circle_theorem#Example :
Complex matrix: [math]\displaystyle{ A = \begin{bmatrix} 10 -1 0 & 1\\ 0.2 8 0.2 & 0.2\\ 1 1 2 & 1\\ -1 -1 -1 & -11\\ \end{bmatrix}. }[/math]
Complex plane regions: [math]\displaystyle{ D(10,2), \; D(8,0.6), \; D(2,3), \; \text{and} \; D(-11,3). }[/math]
- https://en.wikipedia.org/wiki/Gershgorin_circle_theorem#Example :
- Counter-Example(s):
- See: Semyon Aronovich Gershgorin, Mathematics, Eigenvalues And Eigenvectors, Matrix (Mathematics), Doubly Stochastic Matrix, Hurwitz Matrix, Metzler Matrix.
References
2021
- (Wolfram MathWorld, 2021) ⇒ Eric W. Weisstein (1999-2021). "Gershgorin Circle Theorem"]. From MathWorld--A Wolfram Web Resource. Retrieved: 2021-01-23.
- QUOTE: The Gershgorin circle theorem (where "Gershgorin" is sometimes also spelled "Gersgorin" or "Gerschgorin") identifies a region in the complex plane that contains all the eigenvalues of a complex square matrix. For an $n\times n$ matrix \mathbf{A}, define
- QUOTE: The Gershgorin circle theorem (where "Gershgorin" is sometimes also spelled "Gersgorin" or "Gerschgorin") identifies a region in the complex plane that contains all the eigenvalues of a complex square matrix. For an $n\times n$ matrix \mathbf{A}, define
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$R_i=\displaystyle \sum_{j=1, i \ne j}^n \left|a_{ij}\right|$ |
(1) |
- Then each eigenvalue of $\mathbf{A}$ is in at least one of the disks
- Then each eigenvalue of $\mathbf{A}$ is in at least one of the disks
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$\{z:\left|z-a_{ii}\right|\leq R_i\}$ |
(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
- 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
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$\{z:\left|z-a_{jj}\right|\leq S_j^{(r-1)}\}$, |
(3) |
- or in one of the regions
- or in one of the regions
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$\{z:\displaystyle\sum_{i\in P} \left|z-a_{ii}\right| \leq \displaystyle\sum_{i \in P}R_i\}$ |
(4) |
- where $P$ is any subset of $\{1,2,\cdots,n\}$ such that $\left|P\right|=r$ (Brualdi and Mellendorf 1994).
2019
- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Gershgorin_circle_theorem Retrieved:2019-7-28.
- In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hirschhorn.
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.