# 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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### 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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**2**answers

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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 $D$ ...

**2**

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**1**answer

165 views

### 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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### Spectral radius of multivariable operators

Let $E$ be an infinite-dimensional complex Hilbert space.
The spectral radius of a commuting multivariable operator $A = (A_1,\cdots,A_n)\in\mathcal{L}(E)^n$ (i.e. $A_iA_j=A_jA_i$ for all $i,j$) is ...

**3**

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**1**answer

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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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**1**answer

154 views

### 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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**1**answer

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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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**1**answer

436 views

### 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 ...

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**1**answer

519 views

### A spectral radius inequality

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) ...

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**2**answers

651 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 ...