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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Bound spectral radius after multiplied by a zero-row-sum matrix

I'm trying to bound the spectral radius of a matrix product. The first matrix is $$H = (I-M)^{-1}\Gamma,$$ where $\sum_jM_{ij}<1$, $M_{ij}\geq0$, and $\Gamma$ is a diagonal matrix with non-negative ...
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Matrix with spectral radius 1 that converges

Let $\mathbf{A}$ be a matrix of size $n \times m$ with $\sum_{j = 1}^m A_{ij} = 1$ and $\mathbf{E}$ be a matrix of size $n \times m$ with $\sum_{i = 1}^n E_{ij} = 1$ and $\forall i,j \ \ \ \ 1 \geq ...
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For which values of $\alpha$ and $\beta$ is the matrix $A$ strictly diagonally dominant?

Let we have the following matrix: $$ \begin{matrix} 1 & \alpha & 0 \\ \beta & 1 & 1 \\ 0 & 1 & 1 \\ \end{matrix} $$ Is the matrix $A$ strictly diagonally ...
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The spectral radius of a $n\times n$ matrix

I would like to know which is the spectral radius of this $n\times n$ matrix: $$ \begin{matrix} 0 & 1 & . & . & . &1 \\ 1 & 0 & . & . & . &0 \\ ....
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23 views

Compute the value of a series of integrals taking values in a Banach space

I have a question regarding this post that I wrote some time ago: Question about a proof regarding the spectral radius of a linear bounded operator At the end of the question I wrote that: by ...
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36 views

Eigenvalues of the sum of a symmetric matrix $A$ and a hermitian matrix $B$

If I have two matrices: $A \in \mathbb C$ symmetric, and $B \in \mathbb C$ hermitian. Question 1: If the spectral radius of $A$ is much larger than the spectral radius of $B$, can I say that the ...
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Spectrum of operator's module

Let $T : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a translation operator: $$(Tx)(t) = x(t + 1), \quad \forall t \in \mathbb{R}, \, \forall x \in L^2(\mathbb{R}).$$ Find spectrum of operator $\lvert ...
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the convergence of the infinite product of matrices composed by matrces with the spectral radius less than 1

Suppose the two matrices $\mathbf{A}$ and $\mathbf{B}$ have the spectral radius less than $1$. Do the infinite product $\mathbf{C}\left( 1 \right)\mathbf{C}\left( 2 \right)\cdots $ , where each $\...
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27 views

Invertibility of an element in a Banach algebra (Gelfand's formula)

In Folland's A Course in Abstract Harmonic Analysis, Theorem 1.8 states that for a unital Banach algebra (with unit $e$), the spectral radius of an element $x$ is given by $\lim_{n \to \infty} \|x^n\|^...
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Spectral radius of $B$ if $W-B^TWB$ is positive definite

Problem: Suppose that $W = S^TS$ for some square matrix $S$, and that $W-B^TWB$ is positive definite. Show that the Spectral Radius of $B$ is less than $1$. Attempt: $W = S^TS$ is symmetric, so ...
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Prove spectral radius of a primitive matrix is 1 [duplicate]

Let $P \in M (n \times n, \mathbb{R})$ be a primitive matrix. $1$ is a eigenvalue of $P$ and $(1,\dots,1)$ is the associated right eigenvector. How can show that the spectral radius $\rho(P):=$max$\{...
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Bounding 2-norm of powers of a matrix

Suppose that $A$ is a $n \times n$ matrix with $\rho(A) \leq 1$ and $\|A\|_2 \leq R$, where $R>1$. How can I show an upper bound on $\|A^k\|_2$ that is polynomial in $k$? A trivial upper bound is ...
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48 views

Joint Spectral Radius Relation

Let $\theta: \mathbb{N} \rightarrow \Sigma$ a switching signal, $\Sigma=\{1,\dots,m\}$ where $m$ is an integer and $ m \ge 2$, and let $\mathcal A=\{A_\sigma \in \mathbb{R}^{n\times n} |\sigma \in \...
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Spectral radius of a matrix product

Let $A, B \in \mathbb{R}^{n\times n}$ be two invertible matrices. I want to know if one can always find real scalars $\lambda_1, \ldots,\lambda_p$ such that $$\rho\left(\prod_{i=1}^{p} (A- \lambda_i ...
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Gauss-Seidel iteration method Convergence

I was hoping if someone can check my work on my proof on convergence of the Gauss-Seidel method. My friend and I are working on it and we have slightly different answer because of the addition and ...
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1answer
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Eigenvalues of matrix product $ADA^T$

I am considering such a matrix product, namely $ADA^T$, where $A$ is an $n\times m$ matrix with $n>m$, $D$ is an $m\times m$ positive diagonal matrix. I understand the matrix product is positive ...
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44 views

Regarding spectral radius of a matrix

Let $n\geq 2$, $s\leq n$ and $r_1,r_2,\cdots, r_s$ with $\sum_{j=1}^{s}r_j=n. $ Let $E$ be the subspace of ${\mathbb{C}}^{n\times n}$ given by the following scalar block diagonal matrices. $$E= \{diag[...
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spectral radius formula - why does $\sum \xi^n \zeta^{-n}$ converge for $|\zeta| > r(\xi)$?

Every proof of the spectral radius formula $r(\xi) = \limsup \|\xi^n\|^{1/n}$ for Banach algebras (that I have seen) uses the fact, that if $\zeta \in \mathbb{C}$ with $|\zeta| \geq s > r(\xi)$ (...
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$\rho(B) \leq \rho(|B|)$, where $\rho$ is the spectral radius

Let $B_{d \times d}$ be an integral matrix and let $A$ be such that $a_{IJ} = |b_{IJ}|$, $1 \leq I, J \leq d$. Then, $\rho(B) \leq \rho(A)$. My strategy is to use Gelfand's formula, i.e., for a ...
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56 views

Spectral radius of a linear operator the iterates of which are linear combinations

Let $(X,||\cdot||)$ be a real Banach space, let $A_0,A_1:X\to\mathbb R$ be bounded linear functionals and let $v,w\in X$ be fixed. Then for the spectral radius $\mathfrak{R}$ of the operator $A:X\to X$...
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Calculation of a limit, using left eigenvectors, an eigenvector and a positive matrix

I've tried to calculate the limit of 1 over the Spectral Radius of a positive Matrix A times the Matrix A itself, the whole thing to the power of k, but it went wrong somewhere. My attempt is in the ...
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Intuition behind spectral radius

Let $V$ be a normed vector space, and let $T : V \to V$ be a bounded linear operator. Then the spectral radius of $T$, call it $r(T)$ is defined to be $\lim_{n \geq 1} \|T^n\|^\frac{1}{n}$, where $\|\...
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Does spectral radius of a graph increases replacing a subgraph with a graph with more spectral radius?

Let $G$ be a graph with adjacency matrix $A_G$ and $H$ be an induced subgraph of $G$. Let $H^*$ be a graph with same number of vertices as $H$ such that $\rho(A_H) \leq \rho(A_{H^*})$, where $\rho(A)$ ...
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Spectral radius of the product of a block-diagonal matrix and a row stochastic matrix

Let $W\in\mathbb{R}^{N\times N}$ be a right (row) stochastic matrix with non-negative $ij$ entries $w_{ij}\geq0$, where $\sum_{j=1}^N w_{ij} = 1$, and let $A\in\mathbb{R}^{nN\times nN}$ be a block-...
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Question about a proof regarding the spectral radius of a linear bounded operator

I'm studying basic spectral theory from the book Elements of functional analysis by Hirsch and Lacombe, and I've encountered some difficulties in understanding the proof of the following theorem: ...
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83 views

Show that norm of normal operator equals spectral radius via $\| T T^* \| = \| T \|^2 = \| T \|^2$

Let $H$ be a Hilbert spaces and $T \in L(H)$ normal, i.e. $T T^* = T^* T$. Show that $r(T) = \| T \|$, (where $r(T) := \sup_{\lambda \in \sigma(T)} | \lambda |$ is the spectral radius of $T$) by first ...
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35 views

How do i work out if this matrix is diagonally dominted?

I am having trouble answering these two questions: (a) Is matrix A diagonally dominant? (b) Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices.
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sufficient condition on comparing the spectrum of two matrix

I have two square matrix M1 and M2, I wonder what is the sufficient condition for showing the spectrum of M1 is less than M2 (the maximum absolute eigenvalue)
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Are singular values independent of the scalar product?

Let $H$ be a hilbert space $ (\varphi_j)_{j \in \mathbb{N}} \subseteq H$ a Rieszbasis and by $\tilde{\varphi_j}$ we denote the biorthogonal sequence. Meaning $ \langle \varphi_j , \tilde{\varphi_k} \...
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Ratio of largest eigenvalue to largest singular value

How much larger can the largest singular value of a matrix be, relative to the largest eigenvalue? Specifically, given some square matrix $A$ with spectral radius greater than $0$, can one derive a ...
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1answer
38 views

Understanding matrix norms and spectral radius

I thought I understood matrix norms and spectral radius after reading proof of $||A||\geq \rho(A)$. However, in the lecture notes I'm given the following line, and asked what is wrong with the ...
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1answer
79 views

Spectral radius of a matrix multiplied by a contraction matrix

Suppose I have a contraction (or non-expansive) matrix $U \in \mathbb{R}^{n\times n}$, which satisfies $\left\lVert U \right\rVert_2 \leq 1$. Given some matrix $A \in \mathbb{R}^{n \times n}$ can one ...
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1answer
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How to find a sharper lower bound?

$A=\begin{pmatrix} B+T && E\\E^T&& I\end{pmatrix}$ Here $E=\begin{pmatrix} 1& 1& 1& \ldots &1\\ 0& 0& 0& \ldots &0\\ \ldots &\ldots &\ldots&...
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Stuck on the formula of spectral radius

I am having two questions on spectral radius: We denote spectral radius of a matrix $A$ by $\rho(A)$ Question 1: Is it true that for any two matrices $A,B$, $\rho(A+B)\le \rho(A)+\rho(B)$ When ...
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73 views

How to find a tighter upper bound on the spectral radius of the given matrix?

Given that the eigenvalues of this matrix are all real, I need to find the spectral radius of this matrix \begin{pmatrix} 2n-1&& n-1&& n\\ 1&& 2n-3&& 0\\ 1&& ...
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Euclidian Matrix norm vs spectral radius [closed]

I am little bit confused about the difference between the Euclidian matrix norm, spectral norm and and spectral radius. The spectral norm is induced from matrix norm. But what does it mean? When ...
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If $A$ is self-adjoint, how are $\inf\sigma(A)$ and $\inf_{\left\|x\right\|_H=1}\langle Ax,x\rangle_H$ related?

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be self-adjoint, $r(A)$ and $\sigma(A)$ denote the spectral radius and spectrum of $A$, respectively, and \begin{align}\lambda_{\text{min}}&...
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238 views

“Almost Normal” Matrix and Gap between Spectral Radius/Norm

Let's denote $$\Vert{A}\Vert := \max_{x\neq0}\frac{x^* Ax}{x^*x}$$ and let $\rho(A)$ denote the largest absolute value of the eigenvalues of matrix $A$. From basic linear algebra, one could ...
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Prove that the eigenvalue with largest absolute value is positive and real

Let $A \in \mathbb R_{>0}^{n \times n}$. Consider all the eigenvalues of $A$, including complex-valued ones. Prove that the eigenvalue that has the largest absolute value is a positive real ...
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Show that $A^4=A^2$ for an adjacency matrix with spectral radius $\leq1$

Let $\Gamma$ be a finite, undirected graph. Let $A=(a_{ij})$ be its adjacency matrix (that is $a_{ij}=$number of edges from vertex $v_i$ to vertex $v_j$) and thus $A$ is symmetric. Show that if the ...
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Can you help me with a Spectral and Point Spectrum Problem

The Question: Let $T: l^2 \rightarrow l^2 $ be defined by $Tx = T(\zeta_j) = (\alpha_j \zeta_j)$, where $(\alpha_j)$ is dense in $[0,1]$. Find $\sigma_{p}(T)$ and $\sigma(T)$. Here is what I have ...
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59 views

The Volterra operator question in Functional Analysis

Consider $K : C^0[0, 1] → C^0[0, 1]$ given by $K(f)(x) = \int_{0}^{x}f(t) dt.$ Check it is well-defined, linear and continuous. Find $||K||, K(C^0[0, 1]), σ_p(K), σ_c(K)$ and $σ_r(K)$. Also, check the ...
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With basic iterative method $Mx_{k+1} = Nx_k + b$, show that $||x_k-x||_2 \leq \dfrac{\rho(G)}{1-\rho(G)}||x_k-x_{k-1}||_2$

With $Mx_{k+1} = Nx_k + b$, define $G=M^{-1}N$ show that $$||x_k-x||_2 \leq \dfrac{\rho(G)}{1-\rho(G)}||x_k-x_{k-1}||_2$$ where $\rho(G)$ is the spectral radius of G Could anyone help me?
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spectral radius of $G = M^{−1}N$ approximately satisfies $\rho(G) \approx \frac{||x_{k+1}-x_{k}||}{||x_{k}-x_{k-1}||}$

The basic iterative Method to solve linear system $Ax=b$ is: $$Mx_{k+1} = Nx_k+b$$ We define that $G = M^{-1}N$ Show that the spectral radius approximately satisfies $$\rho(G) \approx \frac{||x_{...
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17 views

How to estimate the spectrum of $SN$ if $N$ is a nuclear operator and $S$ is a self-adjoint operator with bounded inverse.

Let $H$ be a hilbert space $S,N: H \to H$, where $S$ is a self adjoint operator and $ A \leq ||S|| \leq B$ for some $A,B > 0$, and $N$ is a nuclear operator with eigenvalues $\lambda_k$. The ...
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1answer
39 views

Bound on spectral radius

I've been struggling with a question. It goes as follows: Given a matrix $A$ $$ \text{s.t. } \rho(A^TA)=1.$$ Prove: $$\rho((I-A^TA)(A^TA)^v)\le \frac{v^v}{(v+1)^{v+1}}$$ I've started thinking on a ...
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1answer
44 views

spectrum of elements in $C^*$ algebra

Suppose $x,y$ are two invertible positive elements in a $C^*$ algebra $A$,if $\|x\|=\|y\|$,can we compute the spectrum $\sigma(x^{-1}y)$ of $x^{-1}y$?Does there exist a relationship between the ...
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1answer
36 views

Why does the spectral radius satisfy $r(A^{2})=r(A)^{2}$?

I am unsure about one direction of the equality: $r(A^{2})=r(A)^{2}$ where $A$ is $n\times n$ dimensional matrix. It is clear that for any Eigenvalue $\lambda$ of $A$, and an associated eigenvector $...
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1answer
88 views

If $A$ is self-adjoint, then $\left\|A\right\|=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle}{\left\|x\right\|^2}$

Let $H$ be a $\mathbb R$-Hilbert space and $A\in\mathfrak L(H)$ be self-adjoint. I want to show that $$\left\|A\right\|_{\mathfrak L(H)}=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle_H}{\left\|...
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42 views

Find Coordinates on circles circumference.

i am having some trouble with some maths that i am dealing with. To be more specific, i have 2 circles : first circle O with center (Xo,Yo) Radius R1 i also have a point A(Xa,Ya) on that circles ...