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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Showing matrix C has spectral radius less than 1

We are given that $A$ is symmetric and positive definite, and $B$ is such that $A-B-B^T$ is also symmetric and positive definite. We are asked to show that $C=-(A-B)^{-1} B$ has spectral radius less ...
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Lipschitz and the spectral radius

let's say $M_{n\times n}$ is a matrix, I know the following theorem: if $Mx+d=x$ has a unique solution ($x^*$) then we have: the sequence $x_0,f(x_0),f(f(x_0)),\dots$ is convergent to $x^*$ for all $...
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Generalizing the interlacing property of eigenvalues for matrix pencil

Let $A_1, \dots, A_k, B_1, \dots, B_k\in\mathbb{R}^{n\times n}$ be symmetric positive definite matrices, and suppose that they all commute with each other. Let $U\in\mathbb{R}^{n\times r}$ be a matrix ...
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Spectral Radius of the Convex Combination of Identity Matrix and a Nonnegative Matrix

Let $D = \operatorname{diag}(d_1, d_2, \ldots, d_n)$ with each of its diagnal entry $d_i \in (0,1)$. $B$ is a non-negative matrix. Consider the following matrix $$ A = DI + (I-D)B, $$ where $I$ is ...
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Showing that the spectrum of the Volterra operators is {0}

I am trying to solve exercise 6, p.231 from Lax's Functional Analysis book. Supposing that $K(s,t)$ is a continuous function of $s,t$ in $t\leq s$, I want to show that $\textbf{K}$ as an operator from ...
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Spectral radius of a matrix.

Suppose I have a matrix $A$ which has the following property $\rho(A)\leq \rho(A_{\rm sym})$, is it true that $\rho(DAD)\leq \rho(DA_{\rm sym}D)$ for a positive definite diagonal matrix $D$?
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Derivative of Spectral Radius of Matrix $\exp(A(t))$

I am faced with the practical problem of solving a system $$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$. ...
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If $\langle A_kx,y\rangle$ is bounded for all fixed $x,y$, does it follow that $\Vert A_k\Vert$ is bounded?

Let $A_1, A_2, \ldots$ be a sequence of finite-dimensional linear transformations. Assume that, for each fixed $x$ and $y$, the sequence $\langle A_kx,y\rangle$ is bounded. Does it follow that $\Vert ...
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Determining the Radius of a Sphere [closed]

During a Physics and Chemistry experiment with students in a 12th-grade class, the teacher has a cylindrical container with a radius of 120 mm, containing water to a height of 50 mm. The teacher ...
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Spectral Radius at the Unique and Globally Attractive Fixed Point of a Specific Type of Mapping

Consider a mapping $f: \mathbb{R}_+^n \rightarrow \mathbb{R}_{++}^n$. If it is monotonic and strictly subhomogeous, then it is contractive under the Thompson’s metric (See Lemma 2.1.7). Here, ...
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Spectral radius of $\sum_{i=1}^N (A_i \oplus B_i) \otimes (A_i^* \oplus B_i^*)$ vs $\sum_{i=1}^N A_i \otimes A_i^*$,$\sum_{i=1}^N B_i \otimes B_i^*$

Consider a set of $n$ by $n$ matrices $A_i$ and a set of $m$ by $m$ matrices $B_i$, with $i=1,2,...,N$. I'm interested in the following operators: $$O_A = \sum_{i=1}^N A_i \otimes A_i^*$$ $$O_B = \...
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Spectral radius of $\sum_{i=1}^N U_i \otimes J_i \otimes U^*_i \otimes J^*_i$ bounded by that of $\sum_{i=1}^N J_i \otimes J^*_i$ for unitary $U$

Consider a set of $n$ by $n$ unitary matrices $U_i$ and a set of arbitrary $m$ by $m$ matrices $J_i$, with $i=1,...,N$. Consider the eigenvalues $\{ \lambda_{U \otimes J} \}$ of the operator $O_{U \...
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Spectral radius of product of matrices with same left and right eigenvector

Suppose $A, B\in \mathbf{R}^{n\times n}$ where $A\mathbf{1}=B\mathbf{1}=\mathbf{1}$, $\mathbf{w}^TA=\mathbf{w}^TB=\mathbf{w}^T$, $\mathbf{w}\geq 0$ and $\rho(A)=\rho(B)=1$. Prove that $\rho(AB)=1$?
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Upper bound for Frobenius norm based on spectral radius?

Preliminaries: The Frobenius of a matrix $A \in \mathbb{R}^{n \times n}$ is defined by $$\|A\|_F := \sqrt{\mathrm{tr}(A^TA)}$$ i.e. the usual Euclidean norm of $A$ if we consider $A$ as a vector of ...
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Resolvent function series converges for $\|\lambda\| \geq r(a)$, the spectral radius

Let $\mathcal{A}$ be a unital commutative Banach algebra over the complex numbers. Fix an element $a \in \mathcal{A}$. I am working with the resolvent set $\{\lambda \in \mathbb{C}: \lambda 1_{\...
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How to prove the inequality for the spectral radius and spectral norm of an arbitrary matrix?

How to prove $\rho(A)\leq \min_D||D^{-1} A D||_2$, where $D$ is any invertible matrix, $\rho(\cdot)$ is the spectral radius, $\||\cdot\||$ is the spectral norm for the matrix? Are there any other ...
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Bounds on the spectral radius of a directed graph

Suppose $(G_n)$ is a sequence of simple directed graphs with $G_n$ having $n$ edges such that the adjacency matrix $A_n$ of $G_n$ is primitive, and let $(G_n’)$ be a sequence of subgraphs obtained by ...
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Continuous-time Switched Linear Systems

Suppose we have a set of matrices $\Sigma$ and a switching rule $\sigma: \mathbb{R} \to \Sigma$. It is known in the discrete case that if the joint spectral radius, $\rho(\Sigma)$, is less than one, ...
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Please help with proving or disproving the statement related to vector induced norm of real symmetric matrices.

Consider any vector induced norm (need not be just p-norm). I need to prove or disprove that for any real symmetric matrix A and vector induced norm $||.||$, $$ ||A|| = \rho (A) $$ where $\rho(A)$ is ...
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What can we say if the spectral radius of a matrix is less than one?

I am currently studying directed acyclic graphs (DAGs). Recall the spectral radius $r(B)$ of a matrix $B$ is the largest absolute eigenvalue of $B$. The paper I am reading said the condition that $r(B)...
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Why shall the spectral radius of a finite Toeplitz matrix not converge to its infinite counterpart?

I encountered this problem when studying Spectral Properties of bounded Toeplitz Matrices by Bottcher & Grudsky. For each polynomial $a=\sum_{n}a_n t^{n}$ which is in Wiener algebra, we define its ...
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Spectral norm of product and spectral radius

I have been thinking about the following problem on the upper bound of the spectral norm of the product: Consider $||\cdot||$ as the spectral norm, by the definition of matrix norm we have $$||AB||\...
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Spectral norm of a symmetric strictly stochastic matrix on simple random walk

I am considering the spectral norm of a transition matrix on simple random walk om $\mathbb{Z}^2$ as follows. Let $S_n$ be a simple random walk on $\mathbb{Z}^2$. Let $D$ be a connected subset of $\...
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Spectral radius of symmetric matrix with negative entries multiplied by a diagonal matrix

Let $M$ be a irreducible, symmetric matrix with some negative entries such that $M^k>0$ for some $k>k_o$ and $\sum_j m_{ij}=1$ with $m_{ij} \in \Re$, and spectral radius $\rho(M)=1$. After ...
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Largest eigenvalue of the covariance matrix of a smooth unimodal probability distribution on the sphere?

For any probability distribution $P$ on the sphere $\Bbb S^n\subset \Bbb R^{n+1}$ let $\lambda_{max}(P)$ denote the largest eigenvalue of its covariance matrix $$ Cov(P)=\left(\int_{\Bbb S^n} ...
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Largest possible eigenvalue of the covariance matrix of a smooth probability distribution on the sphere is attained for the uniform distribution?

Let $\mathcal P$ the set of all probability distributions on the sphere $\Bbb S^n\subset \Bbb R^{n+1}$ that have smooth densities with respect to the Lebesgue surface measure. For all $P\in\mathcal P$ ...
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Can I bound the spectral radius (or maximum eigenvalue) of a positive definite matrix?

This is a problem motivated by spatial autoregressive models. Assume $A$ is an $n\times n$ non-negative matrix with the sum of each row equals to $1$. Let $I$ be an $n\times n$ identity matrix and $c\...
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How to prove the spectral radius satisfies the strict inequalities

Let $A = a_{i,j}$ be a positive matrix, which not all of whose rows have the same sum. Prove the the spectral radius satisfies the strict inequalities $$\displaystyle{\min_i}\sum_{j=1}^na_{i,j}<𝜌(...
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Is there a notion of the spectrum of $L^1$ functions and does it correspond with the support of a random variable?

Is it possible to talk about the spectrum of a function? I know that $$A:=(L^1(\mathbb R^n,\mathbb C),+,*)$$ is a Banach algebra (the product operation ‘$*$’ is the usual convolution product). ...
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Upper bound for sum of singular values of symmetric hollow matrix

Let $\bf{H}$ be $n\times n$ real symmetric matrix with zero diagonal and entries $h_{ij} \in [0, M]$. Denoting its eigenvalues $\lambda_1 \leq \dots \leq \lambda_n$, I want to improve the upper bound $...
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Proof of $\limsup_{n\to\infty} \|a^n\|^{1/n} \le r(a)$

Theorem $2.2.6$ in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff proves the well-known formula for the spectral radius in a unital Banach algebra, namely $r(a) = \lim_{n\to\...
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Bound on spectral radii

Let $L$ be some linear and compact operator on some Hilbert space. And denote by $L^*$ the adjoint operator. I need to bound for some $c>0$: $$ \left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1 $$ ...
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Upper bound on the spectral radius of a matrix

Let a $p \times p$ matrix $A_p$ be defined as follows $$ [A]_{ij} = \frac{1}{1+i+j} $$ for all $i, j \in \{1,\dots,p\}$. What is its spectral radius, $\rho\left(A_p\right)$? If a closed-form solution ...
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Largest value of $\lambda_1$ in $\frac{h_1}{\lambda_1 - 2h_1} + \frac{\sum_{i=2}^n h_i}{\lambda_1} + \frac{2\sum_{i=2}^n h_i^2}{\lambda_1^2} \le 1.$

Suppose $h$ is a probability distribution over $d$ outcomes, and $\lambda(h)$ is the largest value of $\lambda_1$ satisfying the following inequality: $$\frac{h_1}{\lambda_1 - 2h_1} + \frac{\sum_{i=2}^...
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Find the radical of this two dimensional quotient algebra.

enter image description here My think: \lamda \in \sigma(x) if and only if \lamda e -x is not invertible. if x \in B, \sigma(x) , now y \in rad(B) if \sigma(y) = {0} , so A = [a b : 0 a] eigenvalues ...
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Relationship between graph walks and spectral radius

I want to explore what's the relationship between graph walks and spectral radius. I read this result in a paper, however, I could not either find the material that the paper cite on the internet or ...
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Proving that $\sum_u p_u^2 \leq \lambda^2 + \frac{1}{n}$ for an $(n, d, \lambda)$-graph

I am solving a question that involves two parts. For a probability distribution $\pi$, define the collision probability as $CP(\pi) = \sum_{u}\pi_u^2$. Prove that if a probability distribution $X$ ...
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Sensitivity of spectral radius

Eigenvalues are known to be sensitive to the perturbation in a matrix element. Is the spectral radius of a matrix somehow less sensitive?
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Does $\sup_{\Vert x \rVert = 1} \langle A x, x \rangle$ equal the largest eigenvalue of $A$?

If $A$ is a $n \times n$ matrix, taking $x$ to be an eigenvector associated with the largest eigenvalue $\lambda_\max$ yields $\sup_{\Vert x \rVert = 1} \langle A x, x \rangle \ge \lambda_\max$. It is ...
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Estimate with spectral radius and Frobenius inner product

Is the following inequality true for a matrix $C \in \mathbb{R}^{n \times n}$? $$ r(C) \leq \sqrt{|C^{T}:C|}$$ $r(C)$ denotes the spectral radius of $C$ and $X:Y$ denotes the Frobenius-Innerproduct of ...
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Largest eigenvalue of matrix equal to 1

Let $S$ be a correlation matrix with positive entries, show that the largest eigenvalue of $\text{diag}\left(\frac{1}{\sqrt{\boldsymbol{S}\boldsymbol{1}}}\right)\boldsymbol{S}\,\text{diag}\left(\frac{...
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Norm of similar elements in $C^{*}$-algebra - Murphy

I have a question concerning exercise 6 in chapter 2 of Murphy's $C^*$-algebras and Operator Theory. The exercise is as follows: Let A be a unital $ C^* $-algebra. If $r(a) < 1$ and $b = (\sum_{n=0}...
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Character Space of $C^1[0,1]$ and Gelfand Representation

I have recently been working through some exercises in Murphy's "C*-Algebras and Operator Theory," and I am having some trouble with Exercise 10 in Chapter 1. The exercise is as follows: Let ...
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minimizing maximum eigenvalue of a matrix

I have an optimal design problem that tries to minimize the inertia of a robot. To create a robot is to make right choices from a set of candidate parts and mechanisms, so I built a library of those. ...
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How to calculate the spectral radius of $A^TA$ where the matrix $A$ is not symmetric but has same diagonal elements

If the matrix $A$ contains only real values and is also symmetric, then the spectral radius of $A^TA$ is: $$ \rho(A^TA)= (|\lambda|_{max})^2 \tag1$$ where $|\lambda|_{max}$ is the maximum absolute ...
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Is the second norm of matrix $A$ equal to its maximum absolute eigenvalue if $A^TA$ is symmetric?

The second norm of matrix $A$ whose elements are real numbers is the square root of the spectral radius of $A^TA$: $$ ||A||_2= \sqrt{\rho (A^T*A)} \tag1$$ If matrix $A$ is symmetric, its second norm ...
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Estimate spectral radius of general square matrix

I have a large sparse $n \times n$ matrix $A$, which generally is not diagonally dominant. This answer provides an estimate assuming diagonal dominance, but it looks to be inappropriate for the non-...
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Spectral Radius of Perturbed Matrix

Let $A$ be an n-dimensional square matrix such that each entry of $A$ lies in $[0,1]$, i.e., $a_{ij} \in [0,1]$ for all $1\le i,j\le n$. Let $\widetilde{A}$ be a perturbed version of $A$ where some ...
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How can I find the coordinates of the tangent to the arc - JavaScript

I need to find the point of intersection of the tangent to the arc with the frame containing the circle. For this purpose, I create a line tangent to the arc starting at the beginning of the arc and ...
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How to generate square matrices based on constraints on their spectral radii?

After reading the Wikipedia, I am wondering whether there is an algorithm that, given a size $N$, can generate an $N\!\times\! N$ matrix $\boldsymbol{W}$ whose spectral radius satisfies $\rho(\...
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