Questions tagged [spectral-radius]

The spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its spectrum.

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Spectral radius and maximum degree of a graph [duplicate]

How can I show that the spectral radius of a graph $G$ is less than or equal to its maximum degree? I have an unweighted graph $G=(V,E)$, where $V$ is its vertex set and $E$ its edge set. The degree ...
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Spectral radius for block matrix with zero columns

I have a real matrix $$A = \begin{bmatrix} 0 & A_{12} \\ 0 & A_{22} \\ \end{bmatrix}$$ where $A$ is a square matrix with dimension $N \times N$ and $A_{22}$ is also a square matrix with ...
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Size of essential spectrum if $T-\lambda$ is not injective for all $\lambda$ in the essential spectrum.

Let $B$ be some Banach space and let $T:B \to B$ be linear and bounded. I write $\sigma_e$ for the essential spectrum, i.e. the set of $\lambda \in \mathbb{C}$ s.t. $T-\lambda$ is not Fredholm. The ...
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Bounding spectral radius of special matrix (extension of the extension)

This is an extension of Bounding spectral radius of special matrix (extension), which has been already solved. Let $A$ be an $n \times n$ matrix with all nonnegative entries and row sums strictly less ...
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I don't understand our proof for "The spectral radius of a normal operator $A$ is equal to the norm of $A$", it's different to what I've found here

Our proof is a little bit different than the proofs I've found here on this forum. I understand the rest of the proof as it's just using the Gelfand-Beurling formula, but the part I don't understand ...
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Given a family $A$ of $2\times 2$ matrices, $\rho(A)<1$ (JSR) if and only if there is a norm on $\mathbb{R}^2$ s.t each matrix in $A$ is contracting

Given a family $A$ of $2\times 2$ matrices, $\rho(A)<1$ if and only if there is a norm on $\mathbb{R}^2$ s.t each matrix is contracting. I encountered this in a paper and it seems obvious but I am ...
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Spectral radius of a specific matrix multiplied by a diagonal matrix $D=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$

Let $A=(a_{ij})\in M_n(\mathbb C)$, $n\geq 3$ be a matrix satisfying : $\sum_{j=1}^n\lvert a_{ij}\rvert^2=1,$ for all $i=1,\ldots n$. $\sum_{i=1}^n\lvert a_{ij}\rvert^2\leq1,$ for all $j=1,\ldots n$. ...
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Checking a possible corollary of Gelfand's formula

I have recently come across Gelfand's spectral formula for matrices, which Wikipedia gives as follows For any square matrix $A$ and matrix norm, we have $\lim_{k \to \infty} \lVert A^k \rVert ^{\frac{...
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Bounding spectral radius of special matrix (extension)

Let $A$ be an $n \times n$ matrix with all nonnegative entries and row sums strictly less than one, let $V$ be an $n \times n$ nonnegative diagonal matrix satisfying $V \leq I$ (entrywise), let $B\...
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Convergence of matrix with spectral radius 1

The following matrix: $$M = \begin{pmatrix} 0 & \frac{2}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \...
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Bounding spectral radius of special matrix

Let $A$ be an $n \times n$ matrix with all nonnegative entries and row sums strictly less than one, let $V$ be an $n \times n$ nonnegative diagonal matrix satisfying $V \leq I$ (entrywise), let $B\...
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Counterexample of system stability based on theorem: $\rho<1$

A well-known theorem states the following: For any bounded set of matrices $\mathbb{K}$ such that $\hat{\rho}(\mathbb{K})\neq 0$, the joint spectral radius can be defined as \begin{align} \hat{\rho}(\...
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Proving a norm limit based on Schur factorization

I am attempting to do exercises from Trefethen's numerical linear algebra. I am stuck on this one Using Schur decomposition, we are asked to prove that for an arbitrary square matrix $A$ and any ...
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Gelfand's corrolaries counterexample?

Gelfand's corrolaries (https://en.wikipedia.org/wiki/Spectral_radius#Gelfand_corollaries) state that, for any $2$ matrices $\mathbf{A}_1$, $\mathbf{A}_2$, the following relation is true: $ \rho(\...
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how to prove that eigenvalues of a matrix is in unit disk

Define block matrix with real entries $$A=\begin{bmatrix}I-\alpha H-\alpha\beta L &-\beta I \\ \alpha L & I \end{bmatrix}$$ where $H,L$ are real symmetric positive semi-definite (may be ...
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How can I prove that the spectrum of the operator $A\in B(C(K))$ defined by $Af = g∙f$ is equal to $\text{Im}(g)$, for $g\in C(K)$.

I want to prove that $\sigma(A)$, the spectrum of the linear operator $A \in B(C(K))$ which is defined by $Af = g∙f$, is equal to $\text{Im}(g)$, for $g\in C(K)$. We may assume that $K$ is a compact ...
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Is there is any relationship between determinant and spectral radius of the matrix?

We all know that determinant is the product of the eigen values of a matrix. I have found some general term for the determinant of the adjacency matrices from some series of graphs. Can i establish ...
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Spectral radius of the restriction to invariant subspace

Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be two complex Banach spaces such that $X\hookrightarrow Y$ and $X$ is dense in $Y$. Let $T:Y\to Y$ be a bounded linear operator that leaves $X$ ...
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Operators similar to operators with spectral radius 1

Let $A$ be a linear bounded operator acting on a Banach space $X.$ Assume the spectral radius of $A$ is equal $1.$ Do there exist invertible operators $U_n:X\to X,$ such that $$\|U_n^{-1}AU_n\|<1+{...
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Bounding the norms of the powers of a $2\times 2$ matrix

Let $\|.\|_2$ denote the matrix norm induced from the Euclidian vectornorm and let \begin{align} M=\left(\begin{array}{cc} a+b & -b \\ 1 & 0 \end{array}\right). \end{align} I need to bound $\|...
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Suppose that $u$ is invertible, what can be said about $\operatorname{sp}(u^*u)$? Show $u$ is unitary.

Let $\mathcal{A}$ be a $C^*$-algebra, $u\in\mathcal{U}_{\mathcal{A}}$ with $\|u\|\le 1, \|u^{-1}\|\le 1$. What can be said about $\operatorname{sp}(u^*u)$? Show that $u$ is unitary (i.e., $u^{-1}=u^*$...
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Given two matrices pick $\alpha$ to minimize the spectral radius

Given two 2x2 matrices and the following problem without restrictions, which value of $\alpha$ minimizes the spectral radius (the absolute value of the maximum eigenvalue), of both matrices ($\alpha$ ...
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Finding spectral radius of matrix without computing characteristic polynomial

Let $$A=\begin{pmatrix}3&1&1\\1&2&1\\0&1&2\end{pmatrix}.$$ I am asked to find its spectral radius, i.e., $$\rho(A) = \max \left\{ |\lambda| : \lambda \text{ eigenvalue of }A \...
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Rotate 3d shape in path

It seems very simple but I am not able to figure it out let's say I have Plane (yellow) it needs be in border when moved with BLUE direction arrow I have value that needs to be moved for example ...
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Expressing the spectral radius as a function of the matrix size

Currently, I am working with a (non-symmetric) discretization matrix $A$ coming from a convection-diffusion problem without elimination of boundary conditions. Regarding the SOR algorithm, I have to ...
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Strict inclusion of eigenvalues relating to spectral mapping theorem

I've been working on the spectral mapping theorem and I've come across a hurdle while trying to solve a problem. Let P(x) be a non-constant polynomial, T be a bounded linear operator and denote $\...
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Non-diagonalizable 2x2 matrix norm strictly greater than spectral radius

I've seen many answers to problems similar to this, yet I couldn't solve it. Let $J= \mathbb{C}^{2x2}$ be the matrix: $$J = \begin{pmatrix}\lambda & 1\\ 0 & \lambda\end{pmatrix}$$ Show that ...
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Is an eigenvalue always less than or equal to a matrix norm if you drop the sub-multiplicativity requirement, which seems to be optional?

Wikipedia defines a matrix norm as follows: Let $\alpha$ be a scalar in $\Bbb K$ and $A,B$ be $m\times n$ matrices over the field $\Bbb K$. Then a matrix norm $\|\cdot\|$ must satisfy the following ...
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2 votes
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Spectral radius of a graph

I researched about the spectral radius and was confused. There are two definitions. The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix. the spectral ...
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How to prove a positive matrix's spectral radius strictly increases as its dimension increases?

Consider a positive matrix sequence $\{M_1, M_2, \cdots, M_k\}$ where $M_{1} = \begin{bmatrix}x_{1, 1}\end{bmatrix}$, $M_2 = \begin{bmatrix} x_{1, 1} & x_{1, 2} \\ x_{2, 1} & x_{2, 2}\end{...
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Operators with connected spectrum

Let $E$ be a Hilbert space and let $A:E\to E$ be a symmetric bounded operator. The operator $A$ is then self-adjoint. Its essential spectrum $\sigma_{ess}(A)$ is then a closed subset of $R.$ My ...
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On two different spectral radius bounds

Let $\rho(A)$ denote the spectral radius, i.e. $\max_i |\lambda_i|$ with $\lambda_i$ being the eigenvalues of $A$. The bound $(\rho(A))^k\leq ||A^k||_2$ leads to: $$ \rho(A)\leq \tau^\frac{1}{k}\quad\...
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How many integer matrices have spectral radius bounded by a constant?

How many integer matrices with the spectral radius bounded by a fixed constant are there? Without any restrictions the answer is infinitely many. Indeed, $nJ_0$, where $J_0$ is the Jordan cell with ...
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How does the 'reverse' implication follow here?

I am totally confused with the following proof given in Simmons' Introduction to Topology and Modern Analysis book (page: 312): Screenshot Suppose $x$ is an arbitrary element in an Banach algebra and $...
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Spectral radius of discrete Laplacian

We discretize the domain $\Omega = [0,1]^2$ with the step size $\Delta x = \frac{1}{n-1}$. Then, for the largest eigenvalue of the discrete Laplacian it holds $$\lambda_{\max}(-\Delta)\leq 8(n-1)^2.$$ ...
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Bound spectral radius by diagonal elements of diagonally dominant matrix?

Consider a diagonally dominant matrix $A$ with all positive entries. Is it true that the spectral radius of $A$ is lower than or equal to the maximum diagonal entry of $A$?
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Proof spectral radius less than $1$

Given that matrix $A$ is set as: $A=(I-P)(I-QP)^{-1}$ where matrix $QP$ is non-negative reducible hollow matrix, and $\rho(QP)<1$. Matrix $P$ is a diagonal matrix and all entries in $P$ are in ...
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$\|A\|_*=\rho(A)$ for some induced matrix norm when $A$ is diagonalizable.

I'm trying to show that for $A\in \mathbb{C}^{n\times n}$ with $A$ diagonalizable, that there exists an induced matrix norm such that $\|A\|_*=\rho(A)$. I know that $\rho(A) \leq \|A \|$, for all ...
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Finding the Reproduction Number and a final epidemic size in an agent-based network model

I have the following ODE from a disease pandemic in a network ($i=1,2,..., n$): \begin{equation} \dot{x_i} = \beta s_i(1 -l_i)\sum_{j\in N} [A_{ij} (1-l_j)x_j] -(\gamma +\kappa) x_i \end{equation} \...
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3 votes
1 answer
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Show $A^k$ doesn't converge to $0$

Let $A \in \mathbb C^{n \times n}$. Let $r=\{\max \lvert \lambda \rvert \text{ such that }\lambda \in \mathbb C \text { is an eigenvalue of } A\}$. If $r\geq 1$ show that $A^k$ doesn't converge to the ...
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How to show spectral radius of a special matrix is weakly lower than one?

Let $A$ be a strictly substochastic matrix, let $x$ be a scalar with $x\in[0,1]$, and let $$\Lambda(x)\equiv(I-xA^{T})^{-1}.$$ Let $v$ be a vector satisfying $v_{i}\in[0,1],\forall i$ and $\sum_{i}v_{...
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What are different strategies for showing that the spectral radius of a matrix is less than one?

I know I can bound the spectral radius of a matrix by a norm of the matrix. Are there other approaches? In my case, I have a positive square matrix J(v) that depends on some fixed parameters and the ...
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Lower bound for the difference of matrix spectral radii

Let $A$ and $B$ be two nonnegative, symmetric, irreducible matrices with the spectral radii $r(A)$ and $r(B)$, respectively. Show that $r(A)-r(B) \ge x^t (A-B)x$. I am really stuck at this point. I ...
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Understanding spectral radius of sum of two special matrices

For some vector $v$ in the simplex (i.e., $v_k \geq 0, \forall k$ and $\sum_k v_k = 1$) and a matrix $A$ with all entries strictly positive, consider $$J(v) = J_1(v) + J_2(v),$$ with $$J_1(v) = (\...
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Spectral radius of matrix bounded by one

Consider a stochastic vector $v$ ($v_i \geq 0$ and $\sum_i v_i = 1$), a strictly substochastic matrix $A$ ($a_{ij} \in [0,1]$, $\sum_j a_{ij} < 1$), and a substochastic diagonal matrix $D$ ($D_{ij} ...
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Perron-Frobenius Theorem poof by Brouwer fixed point

Could you suggest me a book where I can find a proof of Perron-Frobenius theorem (especially for nonnegative matrices) based on a Brouwer fixed point theorem?
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Spectral radius of matrix products AB vs. CB for positive definite matrices A, B, C

Suppose we have 3 symmetric, positive definite $n \times n$ matrices $A, B, C$. It is known that $A \geq C$ in the sense that the matrix $A - C$ is a positive semi-definite matrix. Let $D = AB$ and $E ...
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Clarification of a point in the proof of $||| A|||\leq \rho(A)+\epsilon$

I want to prove that given $A\in \cal{M}_n$ and $\epsilon>0$, there exists a matrix norm $||| \cdot |||$ such that $$||| A|||\leq \rho(A)+\epsilon$$ where $\rho(A)$ si the spectral radius of $A$. ...
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Non-singular M-matrix Strictly positive Spectral radius

So I've been looking at M-matrices recently, and by its definition we have $A=sI-B$ where $B$ is strictly positive and we have that the spectral radius ($\rho$) (maximum modulus of the eigenvalues )$s\...
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Maximum eigenvalue of a skew symmetric matrix

There is a nice method to find the maximum eigenvalue of a real symmetric matrix: Let $A$ be a real symmetric $n\times n$ matrix. Then the maximum eigenvalue of $A$ is given by, $$\lambda_{\max}=\...
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