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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### 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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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 $r(a) = \lim_{n\to\... 2 votes 1 answer 22 views ### Bound on spectral radii Let$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$: $$\left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1$$ ... 0 votes 0 answers 52 views ### 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 ... 2 votes 1 answer 70 views ### 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: $$\frac{h_1}{\lambda_1 - 2h_1} + \frac{\sum_{i=2}^... 0 votes 0 answers 17 views ### Find the radical of this two dimensional quotient algebra. enter image description here My think: \lamda \in \sigma(x) if and only if \lamda e -x is not invertible. if x \in B, \sigma(x) , now y \in rad(B) if \sigma(y) = {0} , so A = [a b : 0 a] eigenvalues ... 0 votes 0 answers 36 views ### Relationship between graph walks and spectral radius I want to explore what's the relationship between graph walks and spectral radius. I read this result in a paper, however, I could not either find the material that the paper cite on the internet or ... 0 votes 0 answers 15 views ### Proving that \sum_u p_u^2 \leq \lambda^2 + \frac{1}{n} for an (n, d, \lambda)-graph I am solving a question that involves two parts. For a probability distribution \pi, define the collision probability as CP(\pi) = \sum_{u}\pi_u^2. Prove that if a probability distribution X ... 0 votes 0 answers 17 views ### Sensitivity of spectral radius Eigenvalues are known to be sensitive to the perturbation in a matrix element. Is the spectral radius of a matrix somehow less sensitive? 0 votes 0 answers 38 views ### Does \sup_{\Vert x \rVert = 1} \langle A x, x \rangle equal the largest eigenvalue of A? If A is a n \times n matrix, taking x to be an eigenvector associated with the largest eigenvalue \lambda_\max yields \sup_{\Vert x \rVert = 1} \langle A x, x \rangle \ge \lambda_\max. It is ... 0 votes 1 answer 40 views ### Estimate with spectral radius and Frobenius inner product Is the following inequality true for a matrix C \in \mathbb{R}^{n \times n}?$$ r(C) \leq \sqrt{|C^{T}:C|}$$r(C) denotes the spectral radius of C and X:Y denotes the Frobenius-Innerproduct of ... 0 votes 1 answer 139 views ### Largest eigenvalue of matrix equal to 1 Let S be a correlation matrix with positive entries, show that the largest eigenvalue of \text{diag}\left(\frac{1}{\sqrt{\boldsymbol{S}\boldsymbol{1}}}\right)\boldsymbol{S}\,\text{diag}\left(\frac{... 1 vote 1 answer 67 views ### Norm of similar elements in C^{*}-algebra - Murphy I have a question concerning exercise 6 in chapter 2 of Murphy's C^*-algebras and Operator Theory. The exercise is as follows: Let A be a unital C^* -algebra. If r(a) < 1 and b = (\sum_{n=0}... 4 votes 1 answer 119 views ### Character Space of C^1[0,1] and Gelfand Representation I have recently been working through some exercises in Murphy's "C*-Algebras and Operator Theory," and I am having some trouble with Exercise 10 in Chapter 1. The exercise is as follows: Let ... 3 votes 0 answers 109 views ### minimizing maximum eigenvalue of a matrix I have an optimal design problem that tries to minimize the inertia of a robot. To create a robot is to make right choices from a set of candidate parts and mechanisms, so I built a library of those. ... 0 votes 0 answers 61 views ### How to calculate the spectral radius of A^TA where the matrix A is not symmetric but has same diagonal elements If the matrix A contains only real values and is also symmetric, then the spectral radius of A^TA is:$$ \rho(A^TA)= (|\lambda|_{max})^2 \tag1$$where |\lambda|_{max} is the maximum absolute ... 0 votes 1 answer 61 views ### Is the second norm of matrix A equal to its maximum absolute eigenvalue if A^TA is symmetric? The second norm of matrix A whose elements are real numbers is the square root of the spectral radius of A^TA:$$ ||A||_2= \sqrt{\rho (A^T*A)} \tag1$$If matrix$A$is symmetric, its second norm ... 2 votes 0 answers 34 views ### 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-... 1 vote 1 answer 62 views ### 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 ... 0 votes 0 answers 32 views ### 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 ... 0 votes 1 answer 32 views ### 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(\... 