Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [spectral-radius]

The spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its spectrum.

Let $A$ be an $n\times n$ matrix. Its spectral radius $\rho(A)$ is the largest absolute value of its eigenvalues $\lambda_i$, i.e. $$ \rho(A) = \max_{1\leq i\leq n} |\lambda_i| $$

A closely related term is the spectral norm. These are not necessarily the same: for instance, if $A=\pmatrix{0 & 1\\ 0 & 0}$, then $\rho(A)=0$ and $||A||_2=1$. In general we have $\rho(A)\leq ||A||_k$ and a powerful result known as Gelfand's formula gives $\rho(A) = \lim_{k\to \infty}||A||_k ^{1/k}$. See for these questions.

Questions about the spectral radius should usually contain some combination of the tags , , , , or similar.

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Norm of a symmetric matrix equals spectral radius

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an operator in $\mathbb{R}^n$ given by $x \mapsto Ax$. Prove…
Erik Vesterlund
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Substochastic matrix spectral radius

Let $M$ be a row substochastic matrix, with at least one row having sum less than 1. Also, suppose $M$ is irreducible in the sense of a Markov chain. Is there an easy way to show the largest eigenvalue must be strictly less than 1? I hope that this…
SKS
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Quick question: matrix with norm equal to spectral radius

For $A\in \mathcal{M}_n(\mathbb{C})$, define: the spectral radius $$ \rho(A)=\max\{|\lambda|:\lambda \mbox{ is an eigenvalue of } A\} $$ and the norm $$ \|A\|=\max_{|x|=1}|A(x)| $$ where |.| is the Euclidean norm on $\mathbb{C}^n$. Problem: Find…
lnth
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Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N \neq 0$. Let \begin{align*} E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-NX) < 1\}, \end{align*} where $\rho(\cdot)$…
user1101010
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Spectral radius of the Volterra operator

The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: $$\rho(A)=\lim_{n\rightarrow \infty} \|A^n\|^{1/n}…
Cantor
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Spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d. entries with variance $\frac{\sigma^2}{n}$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$? Here, $\rho(\cdot)$ denotes the largest…
mellow
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Is spectral radius = operator norm for a positive valued matrix?

For any real-valued square matrix with all positive entries, by Perron-Frobenius theory, we have that the matrix has a dominant eigenvalue that is real, positive, and of multiplicity 1. Thus, the spectral radius is equal to the largest eigenvalue.…
Hedonist
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Operators similar to operators with spectral radius 1

Let $A$ be a linear bounded operator acting on a Banach space $X.$ Assume the spectral radius of $A$ is equal $1.$ Do there exist invertible operators $U_n:X\to X,$ such that $$\|U_n^{-1}AU_n\|<1+{1\over n},\quad n\ge 1\ ?$$ I can do it for…
Ryszard Szwarc
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Spectral radius is not matrix norm.

I have seen an example of matrix $$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$$ whose spectral radius is zero therefore the spectral radius is not matrix norm. Why the spectral radius is not matrix norm in this case Is it possible…
Tien
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Spectral radius, second induced norm

In my textbook there are few facts left without any sign of a proof, which really bugs me, and I was thinking maybe someone can help me: $A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = \sqrt{\rho(A^TA)}$ where $\rho(A^TA)$ denotes spectral radius…
xan
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Proving result on spectral radius

How do I prove that $$\rho(A)=\inf\limits_{\text{operator norms}}\|A\|,$$ $\rho$ being the spectral radius, $A$ being a complex $n\times n$ matrix and operator norms being induced from vector norms by defining…
MickG
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Proof of Gelfand's formula without using $\rho(A) < 1$ iff $\lim A^n = 0$

Gelfand's formula states that the spectral radius $\rho(A)$ of a square matrix $A$ satisfies $$\rho(A) = \lim_{n \to \infty} \|A^n\|^{\frac{1}{n}}$$ The standard proof relies on knowing that $\rho(A) < 1$ iff $\lim_{n \rightarrow \infty} A^n = 0$.…
Ng E-Jay
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Spectral radius of the restriction to invariant subspace

Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be two complex Banach spaces such that $X\hookrightarrow Y$ and $X$ is dense in $Y$. Let $T:Y\to Y$ be a bounded linear operator that leaves $X$ invariant, i.e. $T(X)\subset X$. Furthermore, suppose…
Jake28
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"Almost Normal" Matrix and Gap between Spectral Radius/Norm

Let's denote $$\Vert{A}\Vert := \max_{x\neq0}\frac{x^* Ax}{x^*x}$$ and let $\rho(A)$ denote the largest absolute value of the eigenvalues of matrix $A$. From basic linear algebra, one could characterize normal matrices as those unitarily…
Yu-Hsuan Huang
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Max eigenvalue of symmetric matrix and its relation to diagonal values

I saw few questions about it, but still can't understand. Let $A$ be a symmetric matrix and $\lambda_{\max}$ its largest eigenvalue. Is the following true for all $A$? $$ \lambda_{\max} \ge a_{ii} \forall i $$ That is, is the largest eigenvalue of…
ChaosPredictor
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