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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Problem in Gershgorin circle theorem for identity matrix

I want to apply the Gershgorin circle theorem in a specific matrix (the identity matrix) $$A=\begin{bmatrix} 1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ 0&0&1&0&...
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Gershgorin-type bounds for smallest eigenvalue of positive-definite matrix

I would like to know if there are known results for bounding eigenvalues of positive-definite matrices, in particular gram matrices $AA^T$ based on easily computable functions of $A$. Gershgorin ...
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Gerschgorin theorem

One tricky question: (a) Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix and let $\tilde{A}=A+E$, where $E\in\mathbb R^{n\times n}$ is such that $e_{ii}=0$ for $i=1,2,\ldots,n$, and $2||E||_{\...
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157 views

Salvaging $A-B$ and $A+B$ invertible imply $A,B$ invertible, for $A,B\in \mathbb{C}^{n\times n}$

Recently, a professor posed the following question on a midterm: If $A,B\in \mathbb{C}^{n\times n}$ and $A-B$, $A+B$ are invertible, prove that $A,B$ are invertible. The question was meant to be, '...
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443 views

How to calculate Gershgorin circles of a matrix with complex entries?

I am trying to approximate or calculate the eigenvalues of this matrix. $$\begin{bmatrix} 4 & 1+i & 0 & 0 \\ 1-i & 3 & -1 & 0 \\ 0 & -1 & -2 & 0.1 ...
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1answer
237 views

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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93 views

How to use Gershgorin disc theorem to determine location of eigenvalues of $M = \begin{bmatrix}I_k&A\\A^T&-I_l\end{bmatrix}$?

There was a quite fast answer to this question regarding invertibility of $M$, but as I discovered more properties, I thought we must be able to find stronger results. $$M = \begin{bmatrix}I_k&A\...
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307 views

Gershgorin circle theorem and similarity transformations

Consider the following problem, that was part of an old exam I am studying for: Let $$ A = \begin{pmatrix} 4 & 0 & 2\\ -2 & 8 & 2\\ 0 & 2 & -4 \end{pmatrix}$$ Using ...
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103 views

About the Gershgorin theorem implying diagonalization

This is rather a simple question. I know that if I have a $n\times n$ matrix and I have $n$ circles, all disjoint with each other, then that matrix is diagonalizable. So, why is that true?
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205 views

Estimating the condition of a symmetric, real matrix using Gershgorin circles

I'm trying to find a way to estimate the condition $\kappa(A)$ of a symmetric, real matrix $A$ based on the norm $\lVert\cdot\rVert_2$ by using the Gershgorin circle theorem. For this, I calculated: $$...
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1answer
127 views

Shifted eigenvalues and Gershgorin theorem

Suppose we have a $n\times n$ symmetric positive semi-definite matrix $\mathbf{A}$. Based on Gershgorin circles theorem all the eigenvalues of the, $\mathbf{A}=[a_{ij}]$, are located in the union of $...
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How do eigenvalues change if we perturbe the diagonal entries of a matrix?

Suppose $A \in M_n(\mathbb R)$ is a stable matrix, i.e., all eigenvalues are on the left open half plane of $\mathbb C$. If in particular, all the Gershgorin disks $\Gamma_j$ corresponding to rows $j=...
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307 views

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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192 views

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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110 views

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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427 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 ...
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148 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 ...
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231 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 +...
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260 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 ...
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295 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....
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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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1answer
618 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 $A$ ...
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270 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 &...
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41 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 ...
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734 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, ...
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34 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.
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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......
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131 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{...
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1answer
82 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_{...
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278 views

Prove that diagonally dominant matrices are regular - Reference request

I know that it is easy to prove that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was ...
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552 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 ...
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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 ...
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201 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 ...
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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 ...
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322 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)...
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349 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(...
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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 ...
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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 ...
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1answer
848 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 ...