Mathematics.

matrix theory

Spectral Radius

Linear Algebra40 minDifficulty7 out of 10

You should know: eigenvalues and eigenvectors, matrix norms

Overview

The spectral radius ρ(A) of a matrix A is the largest absolute value among all eigenvalues. It governs the long-run behavior of matrix powers: A^n → 0 iff ρ(A) < 1. The spectral radius satisfies ρ(A) ≤ ||A|| for any matrix norm, and equals the infimum of all consistent norms. It is fundamental to the convergence analysis of iterative methods for linear systems.

Intuition

Raise a matrix to a high power: the dominant behavior is controlled by the eigenvalue with the largest absolute value. If that eigenvalue has |λ| < 1, repeated multiplication shrinks all vectors to zero. If |λ| > 1, the iteration blows up. The spectral radius is the threshold — it tells you whether a matrix iteration will converge, diverge, or oscillate.

Formal Definition

Definition

For a square matrix A with eigenvalues λ_1, ..., λ_n (counted with multiplicity):

ρ(A)=max⁡1≤i≤n∣λi∣\rho(A) = \max_{1 \leq i \leq n} |\lambda_i|

Maximum absolute value of the eigenvalues of A.

Spectral radius
ρ(A)=lim⁡k→∞∥Ak∥1/k(for any consistent norm)\rho(A) = \lim_{k \to \infty} \|A^k\|^{1/k} \quad (\text{for any consistent norm})
Gelfand's formula
ρ(A)≤∥A∥(for any sub-multiplicative matrix norm)\rho(A) \leq \|A\| \quad (\text{for any sub-multiplicative matrix norm})
Norm lower bound

Notation

NotationMeaning
ρ(A)\rho(A)Spectral radius of A
σ(A)\sigma(A)Spectrum of A: the set of all eigenvalues

Properties

Power convergence

An→0 as n→∞  ⟺  ρ(A)<1.A^n \to 0 \text{ as } n \to \infty \iff \rho(A) < 1.

Gelfand formula

ρ(A)=lim⁡k→∞∥Ak∥1/k for any consistent norm.\rho(A) = \lim_{k\to\infty} \|A^k\|^{1/k} \text{ for any consistent norm.}

Norm inequality

ρ(A)≤∥A∥ for any sub-multiplicative norm.\rho(A) \leq \|A\| \text{ for any sub-multiplicative norm.}

Normal matrices

For a normal matrix, ρ(A)=∥A∥2 (spectral norm equals spectral radius).\text{For a normal matrix, } \rho(A) = \|A\|_2 \text{ (spectral norm equals spectral radius).}

Spectral radius of A^k

ρ(Ak)=ρ(A)k.\rho(A^k) = \rho(A)^k.

Applications

Real-world · Computer Science

PageRank's convergence depends on the spectral radius of the Google matrix being less than 1, ensuring the power iteration converges to the stationary distribution.

Worked Examples

  1. 1

    A is upper triangular, so eigenvalues are the diagonal entries: λ_1=2, λ_2=-3.

  2. 2

    Spectral radius is the largest absolute value.

    ρ(A)=max⁡(∣2∣,∣−3∣)=3\rho(A) = \max(|2|, |-3|) = 3

✓ Answer

ρ(A) = 3.

Practice Problems

Mediumfree response

Find the spectral radius of A = [[0,1],[-2,3]].

Mediumfree response

The Jacobi iteration for solving Ax=b uses iteration matrix G = D^{-1}(L+U) where A = D - L - U. State the condition on ρ(G) for convergence, and explain why.

Common Mistakes

Common Mistake

The spectral radius equals ||A||_2.

||A||_2 = σ_max (largest singular value), while ρ(A) = max|λ_i| (largest eigenvalue magnitude). They are equal only for normal matrices.

Quiz

The spectral radius ρ(A) is defined as:
A^k → 0 as k → ∞ if and only if:
Gelfand's formula states that ρ(A) equals:

Summary

  • The spectral radius ρ(A) = max|λ_i| is the largest absolute value among eigenvalues of A.
  • A^k → 0 as k → ∞ if and only if ρ(A) < 1.
  • Gelfand's formula: ρ(A) = lim_{k→∞} ||A^k||^{1/k} for any consistent norm.
  • ρ(A) ≤ ||A|| for any sub-multiplicative norm; equality holds for normal matrices in the spectral norm.
  • Controls convergence of iterative linear solvers (Jacobi, Gauss-Seidel) and stability of discrete dynamical systems.

References

  1. WebsiteWikipedia — Spectral radius