# Questions tagged [gershgorin-sets]

Questions about gershgorin-sets, gershgorin-disks, gershgorin circle theorem, brauer-cassini-ovals, brauer-sets, brualdi-sets, pupkov-solov-sets

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### Proving inequality with absolute values

consider the theorem on root bounds of a polynomial by Fujiwara: For positive numbers $\lambda_1, \lambda_2, \cdots, \lambda_n$ with $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, the zeroes of ...
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### Gershgorin theorem and strictly diagonally dominant matrix

A is a strictly diagonally dominant matrix. Prove $\prod_{i=1}^n (|a_{ii}|-\sum_{j \ne i}|a_{ij}|)\leq |det(A)|$. ps: I tried Gershgorin theorem, but I cannot prove eigenvalues are contained in ...
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### Power Method to Find all Eigenvalues without Shifting

If a given matrix $A$ is known to have three different and real eigenvalues $\lambda_1$, $\lambda_2$ and $\lambda_3$ and we know that they are near $-2$, $2$ and $10$ respectively (using Gershgorin's ...
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### How do the eigenvalues change if we change the diagonal entries of the matrix?

Suppose $A \in M_n(\mathbb R)$ is stable. By stable, we mean the eigenvalues are all on the left open half plane of $\mathbb C$. Now if we decrease the value of $A_{11}$, does the matrix remain stable?...
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### Interval of eigenvalues using Gershgorin circles

We have the matrix $$A=\begin{pmatrix}2 & 0.4 & -0.1 & 0.3 \\ 0.3 & 3 & -0.1 & 0.2 \\ 0 & 0.7 & 3 & 1 \\ 0.2 & 0.1 & 0 & 4\end{pmatrix}$$ We get the row ...
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### Gershgorin circle and complex eigenvalues

As I understood all complex eigenvalue are coming in complex conjugate pairs. Additionally if all the circle don't overlap so in each circle only one eigenvalue exist and if all the circle's centres ...
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### On the supremum norm of matrices

Let $D=diag (d_{ii}) \in M_n(\mathbb R)$ be a diagonal matrix and $E\in M_n(\mathbb R)$ be such that $||E||_\infty < \min _{i\ne j} \Bigg|\dfrac{d_{ii}-d_{jj}}{2}\Bigg|$. Then how to show that ...
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### Gershgorin Circle theorem- implications

(I am considering only real matrices) Does only hold that if the area of all Gershgorin Circles is positiv $\Rightarrow$ the Matrix is positiv definit (trivial) or does also follow the vice versa ...
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### Clarification on Gershgorin's Second Circle theorem

I'm trying to clarify this theorem, in particular a few statements which seem contradictory. 1) If k discs are disjoint from the others, their union contains k eigenvalues. (my lecture notes) and ...
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### Gerschgorin circle theorem question

How can show the following property of Gerschgorins theorem? If $N$ of the disk from a connected domain that is disjoint from the other $m-n$ disk then there are $n$ values eigenvalues of $A$ ...
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