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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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Finding limit of spectral radius to size ratios on a sequence of symmetric matrixes increasing in dimensions

I came across this problem recently: let $r_n$ be the spectral radius of the $n\times n$ matrix $A_n$ defined by ${A_n}_{(i,j)}=n-|i-j|$, find $$\lim_{n\to\infty}\frac{r_n}{n^2}$$ Intuitively, I guess ...
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How does the spectral radius of the Hessian of a box-constrained polynomial with random coefficients change with degree and dimension?

It seems that spectral radius of a symmetric random matrix with i.i.d. entries 0 mean and some assumptions on variance converges to $\sqrt(n)$ where $n$ is the number of dimensions: e.g. https://arxiv....
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Proof that the hausdorff distance $d_H( \sigma (A), \sigma(B)) \le r(A-B)$ for commuting operators $A,B$

I am supposed to prove that for two bounded commuting operators $A, B$ on a banach space, the hausdorff distance of the spectra $d_H(\sigma(A),\sigma(B))$ is less than or equal to the spectral radius ...
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Existence of Limit from Gelfand's Formula

Let $A$ be a square matrix over $\mathbb C$ with spectral radius $\rho(A) > 0$ and $||A||$ denote the operator norm of $A$. By Gelfand's formula, we have $||A^n||^{1/n} \to \rho(A)$ as $n \to \...
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Is it true that the spectral radius is always less or equal to any matrix norm?

The spectral radius of a matrix is the max eigenvalue $\lambda_{max}$ of it.Matrix norm is a kind of norm satisfying some axioms. For any matrix A,is it true that $\lambda_{max}\le \Vert A\Vert$ for ...
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Confusion related to Spectral radius of a matrix

Calculate the spectral radius of $\begin{equation}A= \begin{pmatrix} 1 &1\\ 2 &1 \end{pmatrix} \end{equation} $ My understanding is spectral radius is maximum modulus of the largest eigenvalue ...
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Spectral radius of $I-KH$, where $K$ is a Kalman gain

I have noticed numerically that the spectral radius of $I-KH$, where $K$ is a Kalman gain, is less than or equal to 1. In other words, for some symmetric positive definite matrices $R$ and $C$, and ...
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Understanding Linearity in Spectral Radius: $\rho(\alpha^2(AA^T)^2 - 2\alpha AA^T + I)$ Expression

Let $\rho(A)$ denote the spectral radius of a matrix $A$, defined as $\rho(A) = \underset{i}{\max} |\lambda_i|$, where $\lambda_i$ represents the $i$th eigenvalue of the matrix $A$. In the course of ...
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Convex combination of nonnegative matrices, simplicity of second largest eigenvalue

Let $\lambda_i$ denote the eigenvalues of the matrix $B$ and let $\rho(A)$ and $\rho(B)$ denote the spectral radii of the matrices $A$ and $B$, respectively. Suppose that $A,B$ are nonnegative ...
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How was this bound of the Norm of this Vektor derived? $\|U(x)\|_2^2\le K(x,x)\rho(G_U)$

I was currently reading this article by Robert Schaback and Maryam Pazouki about bases for kernel-based spaces. To ask this question, I'll give a humble Introduction into the tools I'll use. Let $K:\...
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Smallest $k$ such that $|\langle ABx,x\rangle| \le k\langle Ax,x\rangle$, where $A\ge 0$ and $AB$ self-adjoint

Let $A$ and $B$ be linear transformetions of a finite-dimensional inner product space $V$. Let $A$ be positive semi-definite and assume that $AB$ is self-adjoint. Then what is the smallest $k\ge0$ ...
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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 $D$?
Desperado's user avatar
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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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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, ...
maphado fan's user avatar
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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 $\rho(AB)=1$?
Desperado's user avatar
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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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1 vote
2 answers
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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 ...
Sundaresan G's user avatar
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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 ...
Ma Ye's user avatar
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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 $\...
Hoai Nhan 2171 Le's user avatar
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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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1 vote
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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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1 answer
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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 $c>0$: $$ \left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1 $$ ...
newbybay's user avatar
2 votes
1 answer
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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}^...
Yaroslav Bulatov's user avatar
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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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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{...
Philipp123's user avatar
1 vote
1 answer
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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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