# Questions tagged [spectral-radius]

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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### Tail bound on the largest eigenvalue of degree matrix of Gaussian Wigner matrix

Define $D_{-W}$ as a diagonal matrix with $i$-th diagonal element $(D_{[-W]})_{ii}=\sum_{j=1}^n -W_{ij}$.
I have problem on deducing the following result:
$\lambda_{\text{max}}(D_{-W})<\sqrt{(2+\...

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### Tail bound on largest singular value of Gaussian Wigner matrix

I have problem on deducing the following tail bound on largest singular value of Gaussian Wigner matrix
$\|W\|\leq(2+\epsilon)\sqrt{n}$, $\forall\epsilon$, with high probability.
There is a hint: see ...

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### Can we calculate the spectral radius of the universal cover for specific graphs?

Background
For a finite graph $G$, let $\tilde{G}$ denote the universal cover of $G$. For a vertex $v$, let $p_{2n}(v)$ denote the number of paths of length $2n$ that start and end at $v$. The ...

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### If the spectral radius of matrix $A$ is less than $1$, how to construct a positive definite $Q$ such that $Q - A^{H}QA$ is also positive definite?

It is relatively easy to prove that if there exists a positive definite matrix $Q$ such that $Q - A^{H}QA$ is positive definite, where $A^{H}$ means the conjugate transpose of $A$, then the spectral ...

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### Uniform decay of operator norm for smooth family of operators

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...

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### Are these $L_2$-spectral radii approximations strictly increasing whenever $(A_1,\dots,A_r)$ has no non-trivial irreducible subspace?

If $X$ is a complex matrix, then let $\overline{X}=(X^T)^*=(X^*)^T$ (this means that $\overline{X}$ is the matrix obtained by replacing every entry in $X$ with its complex conjugate), and let $\rho(X)$...

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### Fractional reverse direction Cauchy-Schwarz inequality

If $Z_1,\dots,Z_r$ are complex $m\times m$-matrices, then let $\Phi(A_1,\dots,A_r):M_m(\mathbb{C})\rightarrow M_m(\mathbb{C})$ be the linear mapping defined by $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+...

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### Is the rank preserved when the spectral radius is maximized?

If $A$ is a matrix, then let $\rho(A)$ denote the spectral radius of $A$. If $A=(a_{i,j})_{i,j}$, then let $\overline{A}=(\overline{a_{i,j}})_{i,j}$.
Suppose that $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...

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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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### Is $\rho(X_1\dots X_r)^{2/r}\leq \frac{d}{r}\cdot\rho(X_1\otimes X_1+\dots+X_r\otimes X_r)$ for $d\times d$-real matries $X_1,\dots,X_r$?

Let $\rho(A)$ denote the spectral radius of a square matrix $A$. Let $r,d$ be positive integers. Let $X_1,\dots,X_r$ be $d\times d$-real matrices. Then do we necessarily have $$\rho(X_1\dots X_r)^{2/r}...

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### Can the supremum of this quotient of spectral radii be reached?

Let $V$ be a finite dimensional complex inner product space. If $A_1,\dots,A_r\in L(V)$, then define a mapping $\Phi(A_1,\dots,A_r):L(V)\rightarrow L(V)$ by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\...

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### When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality?

Let $V$ be a complex finite dimensional inner product space. If $A_{1},\dots,A_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A_{1},\dots,A_{n}):L(V)\rightarrow L(V)$ be the superoperator ...

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### Largest eigenvalue of matrix A smaller than 1, what about B when A=B+C? [closed]

Suppose I have a square matrix $A$ that only has non-negative real entries and is not symmetric and not primitive either. It has no "special" structure we could exploit. I know that the ...

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### Spectral norm bound for lower triangular matrix

Let $A$ be a $0/1$ square matrix which can be permuted to a non singular or a singular lower triangular matrix. Determinant is either $0$ or $1$. Can we provide tighter upper bounds on its spectral ...

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### Dynamical degree and spectral radius

Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map.
Let's denote $f_{\ast} : \mathrm{NS}(X) \...

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### Essential spectrum under perturbation

Given a Banach space $X$ and a bounded linear operator $T$ on $X$.
It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation.
My question is about minimal ...

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### Projecting onto space of matrices with spectral radius less than one

Consider the space
$$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$
where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times ...

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### How to derive the Euclidean norm of a matrix from its spectral radius [closed]

For a matrix $A\in R^{n \times n}$, if there is a positive $\rho \in (0,1)$ s.t. $\rho(A) \leq \rho$, where $\rho(A)$ represents the spectral radius of $A$.
Can we get the following conclusions:$\Vert ...

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### Strategies for bounding the spectral norm of a tensor?

Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by
$$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$
(...

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### Spectral norm and "operator norm" for hypergraphs

Consider a $d$-regular, $k$-uniform hypergraph: the elements $S$ of its set $E$ of edges are subsets of $V$ of size $k$, and each vertex $v\in V$ is in $d$ edges. We can then define its adjacency ...

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### Spectral properties of finite random matrices with regards to Kurtosis

Given a $N \times M$, $N\ge M$ finite random matrix where the elements are drawn from a probability distribution with Kurtosis $\gamma$.
Is there anything that can be said about the singular values (...

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### Basic question about convergence of top and penultimate eigenvalues of a sequence of operators

$\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\C}{\mathbf C}$
Questions
Let $I$ be the unit interval.
Let $H=L^2(I)$ and $T:H\to H$ be a compact self-adjoint ...

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### A relation between norm and spectral radius for some matrix operators on Banach spaces $\ell^{p}$

Let $A=(a_{i,j})_{i,j=1}^{\infty}$ be a semi-infinite matrix with real entries. Suppose further that $A$ and $A^{T}$ (matrix transpose) represent bounded operators on $\ell^{p}$ for $p\geq1$. Denote ...

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### Proving anti-concentration for the operator norm of a random matrix

If $X$ is a random matrix then I would like to find $\theta >0$ and $\delta \in (0,1)$ s.t I can say,
$$\mathbb{P} \Bigg [ \Big \vert \Vert X \Vert - \mathbb{E} [ \Vert X \Vert ] \Big \vert > \...

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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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### Continuity of local spectral radius

Let $H$ be a complex Hilbert space and let $T \in \mathcal{B}(H)$ be a linear, bounded operator. Given $x \in H$ we define its local spectral radius as $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|...

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### Case of equality in entrywise spectral radius bound

Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that
$$\...

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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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### An absolute value eigenvalue question

Denote by $\circ$ the Kronecker product, let $|\cdot|$ denote the matrix/vector of absolute values, and let $e$ be the vector of all ones. Comparison is entrywise, i.e., $y \ge x$ is equivalent to $...

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### Spectral radius sum of two matrices

Let $W = S + cT$, where $|c| \le 1$ is a real constant and where $S$ and $T$ are square matrices containing real numbers from the interval $[0,1]$.
Assume moreover that the all eigenvalues $\lambda$ ...

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### Joint spectral radius of $\{M,M^T\}$

Let $F$ be a bounded subset of ${\bf M}_n({\mathbb C})$. G.-C. Rota & G. Strang defined the joint spectral radius of $F$ as follows. For $k\ge1$, denote $F_k$ the set of all products of $k$ ...

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### Uniqueness of projection under spectral norm

I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...

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### When does the spectral radius strictly increase?

For bounded linear operators $A$ and $B$ on a Banach space $X$, I'm looking for results which imply that $r(A) < r(A+B)$ (note the strict inequality), where $r(A)$ denotes the spectral radius of $A$...

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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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### Computational complexity for spectral radius of symmetric matrix

What is the best known algorithmic complexity for computing the spectral radius (largest eigenvalue in magnitude, possibly with respect to some precision and confidence) of a symmetric matrix of size $...

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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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### Regarding spectral radius

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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### Expected spectral radius for a sparse Erdős-Rényi binary matrix with a certain density

Let $A$ be an $n \times n$ sparse matrix generated via the Erdős-Rényi method. Here, "sparse" means that $\|A\|_F = O(n)$. I am interested in the relationship between the expectation $\mathbb E(\rho(A)...

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### The largest eigenvalue of a binary matrix with specific density

I would like to find the largest eigenvalue of an $n \times n$ binary matrix of density $p$, i.e., with $p n^{2}$ ones and $(1-p) n^{2}$ zeros. Any idea or reference is welcome.

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### Spectral radius of infinite substochastic upper triangular matrix

Let $M$ be a Markov chain on $\{0, 1, 2, \dots\} \cup \{\delta\}$, where $\Pr(i \to j) > 0$ for $i, j \in \mathbb{N}$ only if $j > i$, and $\Pr(\delta \to \delta) = 1$. This represents a birth-...

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### The spectral radius of a binary matrix - polynomial growth?

(This is a follow-up to The spectral radius of a binary matrix)
Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$.
QUESTION. Is there a $\delta\in\bigl(0,\frac12\...

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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 ...

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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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### Spectral radius of a rank-1 perturbation

Suppose that $\bf A$ is an $n \times n$ matrix, and $\bf u$ and $\bf v$ are vectors. The matrix determinant lemma lets us easily compute the determinant of ${\bf A} + {\bf u} {\bf v}^\top$, while the ...

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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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### optimize spectral radius

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 ...