In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum.

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### Maximum principle for $L_2$-spectral radius approximations

If $X_1,\dots,X_r\in M_d(\mathbb{C})$, then define $\Phi(X_1,\dots,X_r):M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ by $\Phi(X_1,\dots,X_r)(X)=X_1XX_1^*+\dots+X_rXX_r^*$. Let $O_{r,d}$ be the ...
1 vote
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### Graph energy and spectral radius

Suppose $G$ is a simple graph of order $n$ with eigenvalues $\lambda_1\geq \cdots\geq \lambda_n$. I've encountered the quantity $L=\big\vert |\lambda_1|-|\lambda_2|-\cdots-|\lambda_n|\big\vert$. Note ...
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### An inequality for the spectral radius of block matrices

Let $d,m$ be positive integers. Suppose that $A_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$. Let $A$ be the $dm\times dm$ matrix that can be written as a block ...
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### Approximations of the spectral radii of completely positive superoperators

Let $V$ be a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear operators from $V$ to $V$. An operator $\mathcal{E}:L(V)\rightarrow L(V)$ is said to be positive ...
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### About concentration of eigenvalues values of a random symmetric matrix in a specific interval

Given a random symmetric matrix $M$ and two numbers $\lambda_\min$ and $\lambda_\max$ how do we calculate the expected or high probability value of the fraction of its eigenvalues which lie in the ...
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### Survey papers on spectral radius [closed]

Let $M$ be a $n\times n$ matrix. Are there any survey papers which give lower and upper bounds on its spectral radius? What are the ways to find some lower bounds and upper bounds on $\rho(M)$ ...
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1 vote
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### Upper bounds on absolute eigenvalue of sum of two matrix

We have this iteration $$X_{k+1}=(G\cdot Jf+H)X_k+C$$ with $G$ is symmetric and nonnegative, $H$ is nonnegative. $Jf$ is the jacobian matrix of some function $f$ and we can assume it satisfy certain ...
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### Minimize spectral norm under diagonal similarity

Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e., $$s(A) = \min_{D} \lVert D^{-1} A D\rVert_2,$$ where $D$ ...
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Let $A$ be a $C^*$ algebra. Let $a\in A$ be such that $a^*a-aa^*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?
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### The spectral radius of a binary matrix

Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$. It follows from the spectral radius formula that if $M\in\mathcal B_n$ is not nilpotent, then $\rho(M)$, the ...
1 vote
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### best known bounds for spectral radius [closed]

There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the ...
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### Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
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### Operator norm vs spectral radius for positive matrices

I believe the following statement should be true but somehow I don't see an argument: For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
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Hi I would like to solve the following optimization problem. Let $A$ be an $n \times n$ nonnegative real matrix where $A^{-1}$is an M-matrix. Let $D=\text{diag}\{d_{1}, \dots , d_{n}\}$ be a ...
Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) ... 5 votes 2 answers 912 views ### spectral radius monotonicity I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix$A$or$\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
Suppose $F(x)$ is a convex objective function on $n\times n$ matrices, and I need to numerically optimize $F$ with the condition that $x$ has spectral radius less than $1$. This might be too hard, so ...