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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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How to prove Neumann series doesnt converge when spectral radius > 1?

For an operator $T \in B(X)$, its spectral radius is $$r(T) = \lim_{n\rightarrow \infty} \|T^n\|^{1/n}$$ and the Neumann series is $$\sum_{n=0}^{\infty}T^n.$$ If $r(T)>1$, I can show that series ...
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Conditions such that $f(x)= AB^{-1}x$ is a contraction mapping

Consider the function $y = f(x) = AB^{-1}x$, where $x,y$ are vectors, and $A,B$ are matrices. I wish to show that $f(X)$ is a contraction mapping. Question: What are the conditions on $A$ and $B$ ...
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Eigenvalues of dense vs. sparse stochastic matrix

For a stochastic matrix $A$, it is known that the maximum eigenvalue is $1$ and that each eigenvalue $\lambda_i$ satisfies the inequality: $|\lambda_i| \leq 1$. Given a dense (and possibly symmetric)...
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Relationship of spectral radius to matrix norm

I just read that for a real symmetric matrix, the matrix $(A)$ norm equals the spectral radius $(p)$ to the $n^{th}$ power : $||A||=p^n$. I don't think this is true, is it? If so, where does it come ...
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Proof involving the spectral radius and the Jordan canonical form

Let $A$ be a square matrix. Show that if $$\lim_{n \to \infty} A^{n} = 0$$ then $\rho(A) < 1$, where $\rho(A)$ denotes the spectral radius of $A$. Hint: Use the Jordan canonical form. I am ...
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30 views

Change to spectral radius due to removal of a single vertex from graph

Say we have a graph $G$ on $n$ vertices, with eigenvalues $\lambda_1 \leq \dots \leq \lambda_n$ and spectral radius $\rho_G$. Let $H$ be the induced subgraph where we remove a single vertex from $G$, ...
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Suppose a square matrix $A$ has spectral radius $\rho(A) < 1$. Fixing the last row and scaling other entries by $r \in (0,1)$, will $\rho(A)<1$?

Suppose $A \in M_n(\mathbb R)$ has spectral radius small than $1$, i.e., $\rho(A) < 1$. Denote $A = \pmatrix{a_1^T \\ \vdots \\ a_n^T}$, where $a_j^T$ denotes the $j^{th}$ row of $A$.Putting $B=\...
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1answer
33 views

Eigenvalue and the row sum

How could we prove the following statement? Let $A \in \mathbb C^{n \times n}$ such that $\displaystyle \sum_{j=1}^{n} |a_{ij}| \leq 1$ for all $1 \leq i \leq n$, then $\forall \ \lambda \in \...
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2answers
47 views

Spectral Radius $\leq$ min(1-norm, infinity norm)

How do I prove that the spectral radius of a matrix is less than or equal to the minimum of 1-norm and infinity norm of the matrix? i.e. $$\rho(A) \leq min(||A||_1, ||A||_{\infty})$$ I know the ...
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Non-Negative irreducible matrices with random (correlated or independent) non-zero entries and deterministic zeros

Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $\lambda$, is positive and equal to its spectral radius $\rho(M)$. My question is ...
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Richardson's Iteration, Gradient Method and Spectral Radius

Richardson's iteration introduce a scalar $\alpha$ to the update formula: $$ \textbf{x}^{(k+1)} = \textbf{x}^{(k)} + \alpha \textbf{r}^{(k)} $$ And compute $\alpha$ by minimizing the spectral radius:...
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1answer
26 views

Convergence of powers of matrix given convergence of the powers of its absolute value.

I have a matrix A and a matrix B such that $B_{i, j} = |A_{i, j}|$. I am given that all of the eigenvalues of B have magnitude less than 1, and therefore: $ \displaystyle \lim_{k \to \infty} B^k = 0$ ...
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42 views

Matrix norm for two matrices simultaneously close to spectral radius

Suppose $A$ and $B$ have the same spectral radius $\rho$. We can find a norm $\| \cdot \|_A $ s.t. $\|A\|_A - \epsilon < \rho$. We can likewise find a another norm s.t. $\|B\|_B - \epsilon < \...
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Simply find point in circle border

This seems to silly for you expert guys in math but I am not good in this So Please help. Suppose you have clock of 330 * 330 pixels so I have radius 165 in circle. I want to find position of ...
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43 views

Spectral radius equal to 1 and convergence

The following theorem is well-known: $$ \lim_k A^k = 0 \text{ if and only if } \rho(A)<1 $$ (see wiki for context and proofs). What if now $\rho(A)=1$ and $\lambda\neq -1$ for all $\lambda \in ...
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Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1).

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...
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1answer
61 views

Does a matrix of minimum norm in an affine subspace of $M_n(\mathbb R)$ have minimum spectral radius?

Let $\mathcal U \in M_n(\mathbb R)$ be a subspace defined by declaring certain entries to be $0$. More precisely, let $\Lambda, \Theta \subset \{1, \dots, n\}$, $\mathcal U$ is defined as \begin{align*...
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31 views

Does $\lVert B^TA^{-1}B\rVert_2<1$ imply that $\left\lVert\begin{bmatrix}0&-A^{-1}B\\0&-B^TA^{-1}B\end{bmatrix}\right\rVert_2<1$?

Does $\left\lVert B^{\operatorname{T}}A^{-1}B\right\rVert_2<1$ imply that $\left\lVert\begin{bmatrix}0&-A^{-1}B\\0&-B^{\operatorname{T}}A^{-1}B\end{bmatrix}\right\rVert_2<1$, for $A,B\in\...
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1answer
29 views

Finding the spectral radius of operator $ T(x)= (0, x_1, 2x_2, x_3, 2x_4,…)$ on $l^{1}$

My question is:. We have $x=(x_1, x_2,...) \in l^{1}$ and $T: l^{1} \mapsto l^{1}$ is a linear operator defined as $$ T(x)= (0, x_1, 2x_2, x_3, 2x_4,...)$$ then we have to show that--------- 1. The ...
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230 views

What is the limit of $\mathrm{Tr}(G^kM{G^*}^k)^{1/2k}$ when $k$ goes to infinity?

If $G\in \mathscr M_n(\mathbf C)$ then it's well known that $\lim_{k\to \infty}\|G^k\|^{1/k}=\rho(G)$ where $\rho(G)$ is the spectral radius of $G$, the value of the limit does not depend on the ...
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63 views

Prove that $k(A) \geq \rho(A)/\min |\lambda|$ and that $k(A) \geq \rho(A) \rho(A^{-1})$

I'm trying to solve this question here. Thank you in advance for your help. Prove that $k(A) \geq \rho(A)/\min |\lambda|$ and that $k(A) \geq \rho(A) \rho(A^{-1}) $ We assume the matrices $A$ and $...
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1answer
35 views

Spectral radius

Suppose I have a matrix $M$ and $||M||_2$ denotes the spectral radius of the matrix. I came across a note which says $||M||_2 \leq \sqrt{||M||_1||M||_\infty}$. Could someone explain to me how this ...
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1answer
43 views

Geometric series of entries of matrix with spectral radius smaller

I have a matrix $A$ such that the spectral radius is $< 1$. It is well known that $I+A+A^2$... converges. Does it then follow that the geometric series of each entry also converges? The matrix ...
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1answer
55 views

Is it possible to determine the largest number $\tau$ such that the spectral radius $\rho(A\pm \tau ee^T) < 1$

Let $A \in M_n(\mathbb R)$ with no particular structure assumed and the spectral radius $\rho(A) < 1$. Let us denote the all $1$ vector by $e = (1, \dots, 1)^T$. I would like to determine a number $...
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Spectrum of a matrix with normally distributed entries

Let $M \in \mathrm{Mat}_n(\mathbb{R})$ be such that $M_{ij}$ are independent r.v. with distribution $\mathcal{N}(\mu, \sigma^2)$. Than what can we say about $\mathrm{Spec} M$ or at least about $\min\{|...
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1answer
152 views

Fastest converging algorithm for computing spectral radius of symmetric matrix

There are quite a few algorithms available to numerically compute eigenvalues of a matrix. Suppose one is not interested in computing the full spectrum of a general matrix, but only the largest ...
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1answer
77 views

Comparing spectral radii of two non-negative matrices

Let $A$ and $B$ be $n \times n$ non-negative matrices with spectral radii $\lambda_A$ and $\lambda_B$, respectively. Suppose that $A_{ij} \leq B_{ij}$ for all $i,j$. Then, $$\lambda_A \leq \lambda_B$$ ...
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1answer
94 views

Lower bound spectral radius of matrix

Let $\alpha, \beta, \gamma \in (0, 1)$ such that $\alpha < \beta, \gamma < \beta$. Let $d \in \mathbb{N}, d \geq 4$. I am interested in bounding from below the spectral radius $\rho(K)$ of the ...
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Growth of “spectral bounded” sequences of products of matrices

I am trying to proof the following, of which I am sure it is true. I would be pleased if you could give me hints, how to approach such a problem. Given a finite set of matrices $\mathcal{A}=\{A_j\in\...
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1answer
85 views

Rescaling stochastic matrix

Let $A$ be the stochastic transition matrix of an ergodic Markov chain of size $n$ (number of states of the chain). Let $\pi$ be the row vector such that $\pi A = \pi$ (a.k.a. left eigenvector ...
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1answer
154 views

Upper bound of the spectral norm of a matrix power

Let $A\in\mathbb{C}^{n\times n}$ be a complex valued square matrix which can be written as $A=PUP$ in which $P$ is a projector and $U$ is a unitary matrix. The interesting case is $P$ and $U$ do not ...
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2answers
108 views

Quadratic form associated to a linear operator in an Hilbert space

Let $A$ be a linear operator on $H$ Hilbert space (with a scalar product $(\cdot,\cdot)$ ), and let's define $r_{A} =\sup_{\|x\|=1}(Ax, x)$. We want to show that $r_A=\|A\|$, where we endow the space $...
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1answer
43 views

Root test for matrix series via spectral radius?

Let $\sum_{k=1}^\infty A_k$ be a matrix series in $\mathrm{M}_n(\mathbb{C})$ and let $||\cdot||$ be a submultiplicative norm on $\mathrm{M}_n(\mathbb{C})$. Since the root test is valid in any Banach ...
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22 views

Bounds for Spectral Noem

Let $A\in M_n$ and let $\epsilon>0$ be given. Show that there is a nonsingular matrix $C=C(\epsilon)\in M_n$ such that $\rho(A)<|||CAC^{-1}|||<\rho(A)+\epsilon$. I know we need to use Schur'...
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1answer
44 views

Is it possible to upper bound this family of matrices in operator norm?

Let $$\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le \|A_0\|_2\}$$ where $A_0 $ is some fixed matrix and $\|\cdot\|_2$ denotes the induced $2$-norm. We also have for every $A \...
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1answer
79 views

Is there a relation between $\rho(A) + \rho(B)$ and $\rho(A+B)$, where $\rho$ is the spectral radius?

Let $A, B \in \mathcal M_n(\mathbb C)$, we want to know if there's any relation between their spectral radius and the spectral radius of $A + B$. The first part of this exercise is giving examples of ...
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1answer
38 views

Find the spectrum σ(A).

Let A: l2 → l2 be given by A(x1,x2,...)=($\frac{1}{2}$x1,$\frac{2}{3}$x2,...,$\frac{n}{n+1}$xn,...). Find the spectrum $\sigma$(A) and the spectral radius.
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1answer
93 views

The largest eigenvalue of AR(1) matrix

Let $M$ be a $n \times n$ AR(1) matrix whose $(i,j)$-th entry is $$M_{ij} = \rho^{|i-j|}$$ with $0 < \rho < 1$. Is there an explicit formula to compute the largest eigenvalue of $M$?
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Upper bound for spectral radius of matrix multiplications

For Chebyshev iteration, I want to find an upper bound for the highest eigenvalue of a matrix. I have a library in C++ to find eigenvalues for symmetric matrices, but for Chebyshev I need to find the ...
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Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N \neq 0$. Let \begin{align*} E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-...
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1answer
37 views

Example of a matrix where $\rho(A)<\|A\|_p$

It is true that $\rho(A)\le\|A\|_p$ but is there an example where the inequality is strict? ($\rho$ is the spectral radius i.e. $\max|$eigenvalues of $A|$) Though I understand the definition of a ...
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1answer
100 views

Is the Spectral Radius of a Primitive Matrix Bounded?

Let $A$ be a primitive matrix. If I add a nonnegative matrix $\epsilon E$ ($\epsilon > 0$), does the following limit for the spectral radius exist $$\lim_{\epsilon \to \infty} \rho(A+\epsilon E)$$ ...
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2answers
48 views

Checking positive definiteness [closed]

Is the matrix $(A - D^{T}AD)$ positive definite if the spectral radius of $D$, $\rho(D)<1$? Here $A$ is a positive definite matrix and $D^{T}$ denotes the transpose of $D$. $A$ and $D$ are both ...
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1answer
128 views

Spectral radius $\rho\left(A\right)$ of an irreducible, non-negative, and non-singular matrix

Frobenius theorem experts, Why for an irreducible, non-negative, and non-singular matrix $A \in M_n$ (where $n$ is a prime number) is either $\rho\left(A\right)$ is the only eigenvalue of $A$ of ...
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1answer
106 views

Limit of product of matrices with spectral radius less than 1

If spectral radius of a square matrix $A$ is less than 1, then we know that $$\lim_{k \to \infty} A^k = 0.$$ Now, I want to know whether we can also conclude the following results? $$\lim_{k \to \...
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1answer
117 views

Determine whether the set $\{ k \in \mathbb R^{1 \times n} : \rho(A-bk^T) < 1 \}$ is convex

Suppose we have a controllable (stabilizable) pair $(A, b)$ (Definition can be found here and it is equivalent to $(b, Ab, A^2b, \dots, A^n b)$ having full rank), where $A \in \mathbb R^{n \times n}$ ...
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1answer
49 views

The spectral radius is a norm

Let $(A,\|.\|)$ be a complex unital Banach algebra. Let $r(a)$ the spectral radius of an element in $A$. Suppose the spectral radius is a norm on $A$, then does that imply that $\|a\|\leq k r(a)$ for ...
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22 views

Are Cycles good expanders???

The definition of a $k$-regular Ramanujan graph is that $\mu_1$, the largest non-trivial (meaning not equal to $k$) is less than or equal to $2\sqrt{k-1}$. All the eigenvalues of $C_n$ are between $-2$...
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2answers
48 views

Spectral radius of a complete tripartite graph

I am reading an article that mentions that it can be checked that $K_{4,4,12}$ and $K_{2,9,9}$ have the same spectral radius, namely, $12$, i.e., according to the corresponding adjacency matrices with ...
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1answer
63 views

Spectral radius of a complete bipartite graph

Let $p, q > 0$ be integers and let $K_{p,q}$ be a complete bipartite graph. Let $A(K)$ denote the adjacency matrix of $K_{p,q}$ according to a convenient labeling of vertices , which is the $2 \...