Mathematics Stack Exchange
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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.

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319 questions
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Spectral radius of matrix bounded by one

Consider a stochastic vector $v$ ($v_i \geq 0$ and $\sum_i v_i = 1$), a strictly substochastic matrix $A$ ($a_{ij} \in [0,1]$, $\sum_j a_{ij} < 1$), and a substochastic diagonal matrix $D$ ($D_{ij} = 0$ if $i \neq j$ and $D_{ii} \in [0,1]$) . Let $B…
Andres
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Prove spectral radius of a primitive matrix is 1

Let $P \in M (n \times n, \mathbb{R})$ be a primitive matrix. $1$ is a eigenvalue of $P$ and $(1,\dots,1)$ is the associated right eigenvector. How can show that the spectral radius $\rho(P):=$max$\{|\lambda| : \lambda$ is a eigenvalue of $P\}$ of…
Tino
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Proof spectral radius less than $1$

Given that matrix $A$ is set as: $A=(I-P)(I-QP)^{-1}$ where matrix $QP$ is non-negative reducible hollow matrix, and $\rho(QP)<1$. Matrix $P$ is a diagonal matrix and all entries in $P$ are in interval $[0,1]$ I am trying to proof that the spectral…
Simon Liu
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Upper bound of spectral radius of product of matrices

Consider square matrices $A$ and $B$, both of which are inverses of M-matrices, and a nonnegative diagonal matrix D. Is there a nice expression for the upper bound of $\rho \left( D^{-1} A D B \right)$ in terms of matrices $A$ and $B$? If $B=I$…
Andres
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