Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gershgorin-sets]

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

3
votes
1answer
37 views

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 ...
1
vote
1answer
26 views

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 ...
0
votes
0answers
60 views

Gershgorin theorem with complex numbers

I'm having problems with the Gershgorin circle theorem with complex numbers... Let $T:\mathbb{C} \to \mathbb{C}$ be a linear transformation $T(x, y, z, w) = (x+2z, 2αy+2z−w,12x + 3y − 5z, x + y − iz +...
2
votes
1answer
97 views

A theorem related to the Gershgorin's theorem

There is a question/theorem in the book Matrix Analysis by R. Horn, page 351 says that: Let $A \in M_{n}$. Then $\sigma(A) = \bigcap\limits_{S} G(S^{-1}AS)$ if the intersection is taken over all ...
1
vote
2answers
75 views

Gershgorin Circles Theorem: A counterexample?

I have the following matrix: $\begin{bmatrix}8 \ 7 \ 7\\ 0 \ 2 \ \frac14\\ 0 \ 3 \ 1 \end{bmatrix}$ So my Gershgorin Circles are $D(8,14)$ $D(2,0.25)$ $D(1,3)$ The Eigenvalues however are: $8,2....
3
votes
0answers
51 views

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 ...
3
votes
1answer
245 views

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 $A$ ...
2
votes
0answers
143 views

Use the similarity transform $T^{-1}AT$ to show that the matrix $A$ has at least two distinct eigenvalues

Consider the matrices $A$ and $T$ given by $$A=\begin{bmatrix} 3 &\alpha &\beta\\-1 &7 &-1\\0 &0 &5 \end{bmatrix} , T=\begin{bmatrix} 1 &0 &0\\0 &1/2 &0\\0 &...
2
votes
1answer
36 views

Canonical forms which have “minimal” Gershgórin discs, do they exist?

I'm wondering about if there is some way to define, uniquely or not a canonical form which has minimal radii for Gersgórin discs. To be more specific for a given matrix $\bf A$, find $\bf C$ and $\bf ...
2
votes
2answers
392 views

Gershgorin discs and norm of a matrix

Find a matrix, where the estimation of eigenvalues with the help of Gershgorin discs is a, the same as b, worse as the estimation with the help of the norm of the matrix ($||A||_\infty$) So, yes, ...
0
votes
1answer
30 views

Let $S$ be a set with $|S|=n$. So how many subsets $A$ of$ S\times S$ are there with the property that $(a,a)\in A$ $\forall a\in S$

So this is my assignment about subsets.I am absolutely no where with this because i don't have clear idea about subsets.Any kind of help would be appreciated.
4
votes
1answer
935 views

Gershgorin Circle Theorem: counterexample to a statement in the proof?

I have been struggling to comprehend the proof of Gershgorin Circle Theorem for a long time now, but I think I have come upon a counterexample. I'm probably wrong, but please tell me where I'm wrong......
0
votes
2answers
101 views

Gershgorin theorem when a matrix row is $a_{ij} = \delta_{ij}b$

Given a matrix $A \in \mathbb{R}^{N \times N}$ where there is a row (say the $i$-th row) such that $$a_{ij} = \delta_{ij}b ~~~~\forall j \in \{1, \ldots, N\}$$ where $$\delta_{ij} = \left\{ \begin{...
2
votes
1answer
72 views

Finding matrix with given special eigenvalues

Given an arbitrary matrix $A=\left[a_{i,j}\right]\in\mathbb{C}^{2\times2} $ Now define the set $$ \mathcal K = \left\lbrace z\in\mathbb C : \left|z-a_{1,1}\right|\cdot\left|z-a_{2,2}\right|=\left|a_{...
3
votes
2answers
175 views

Proof that diagonally dominant matrices are regular - Reference request

I know that it is easy to proof that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was ...
4
votes
1answer
314 views

Example of a matrix with a gersgorin disk that does not contain any eigenvalue

I am looking for an example of a matrix $A\in\mathbb{C}^{n\times n}$ with the property that at least one Gersgorin Disk $\Gamma_i$ contains no eigenvalue of $A$ for a non-empty proper subset $S$ of ...
3
votes
1answer
908 views

Disjoint Gershgorin disks $\Rightarrow$ each contains exactly one eigenvalue

It is an exercise in Peter Lax's book Linear Algebra that if all the Gershgorin disks $$D_i := \{z\in \mathbb{C} : |a_{ii} - z| \leq \sum_{i \neq j} |a_{ij}|\}$$ are disjoint, then each disk must ...
4
votes
0answers
171 views

Eigenvalues of weighted Laplacian

Let $L_{n \times n}$ be a Laplacian matrix of a directed graph, for example, $$ L = \begin{bmatrix} 2 & -1 & -1\\ 0 & 1 & -1\\ -1 & 0 &1 \end{bmatrix}. $$ Gersgorin disc ...
3
votes
1answer
2k views

Connection between irreducibly diagonally dominant matrices and positive definiteness?

I am attempting to prove a proposition that I found in Cottle's "The Linear Complementarity Problem" book in which the proof has been omitted. I will start by introducing some definitions. ${\bf ...
1
vote
0answers
256 views

Why Gerschgorin Theorem just use the sum of row entries as the radius?

From the Gerschgorin Theorem we know that each eigenvalue located in the circles which radius are:$$r_{i}= \sum_{j=1,j\ne i}^n | a_{ij}|$$(which means add each entries in a row except for the diagonal)...
0
votes
1answer
165 views

What does “the Gershgorin discs $C_j$ corresponding to the columns of $A$” mean?

It says in Wikipedia: Corollary: The eigenvalues of $A$ must also lie within the Gershgorin discs $C_j$ corresponding to the columns of $A$. Before that, a Gershgorin disc is defined as. $D(...
6
votes
2answers
730 views

Is there a version of the Gershgorin circle theorem that is suitable for nearly triangular matricies?

The Gershgorin circle theorem, http://en.wikipedia.org/wiki/Gershgorin_circle_theorem, gives bounds on the eigenvalues of a square matrix, and works well for nearly diagonal matrices. For a ...
4
votes
1answer
3k views

The use of Gershgorin Circle Theorem

In order to estimate the eigenvalues of a real symmetric $n\times n$ matrix, I intend to use the Gershgorin Circle Theorem. Unfortunately, the examples one might find on the internet are a bit ...
2
votes
1answer
593 views

Proving a Theorem based on Gerschgorin Theorem

Here is the theorem as it appears in my textbook. I am so lost with it. For $A=(a_{ij}) \in \mathbb C^{n\times n}$ we have $$\rho(A) \leq \max_i\sum_j^n | a_{ij}|$$ where $\rho(A)$ is the spectral ...