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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### 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 ($i=1,2,..., n$): $$\dot{x_i} = \beta s_i(1 -l_i)\sum_{j\in N} [A_{ij} (1-l_j)x_j] -(\gamma +\kappa) x_i$$ \...
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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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