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A note on the Joint Spectral Radius
Antonio Cicone
TL;DR
The paper examines how to compute and characterize the joint spectral radius when analyzing the robustness of a stable discrete-time system with a family of matrices. It develops properties of the joint spectral radius, including norm independence and spectral-radius equalities for bounded families, while addressing the failure of a finiteness conjecture.
Problem
Analyzing robustness for a stable discrete-time system with a family of matrices leads to the problem of computing the joint spectral radius.
Method
The paper studies joint-spectral-radius properties for bounded matrix families, including norm-based definitions, spectral-radius equalities, and structured-family characterizations.
Results
The joint spectral radius does not depend on the chosen matrix norm, and the Complete Spectral Radius Theorem gives equalities for bounded families.
Takeaways & Limitations
The joint spectral radius provides a basis for analyzing robustness of discrete-time systems governed by matrix families.
Takeaways & Limitations
A conjecture concerning rational matrix families and spectrum-maximizing products is false, so the associated finiteness claim does not hold generally.
Abstract
from arXiv · showhide
A brief summary on the properties of the so called Joint Spectral Radius
1 Terminology, notation and basic properties
This section establishes matrix terminology, spectral concepts, canonical forms, norms, stochasticity, and related notation used throughout the paper.
- Norms and decompositions: Induced matrix norms are submultiplicative, satisfying ∥AB∥_* ≤ ∥A∥_*∥B∥_* for square matrices A and B.The section also introduces vector p-norms, ellipsoidal norms from Hermitian positive definite matrices, and singular value decomposition A = UΛV∗.
- Spectral concepts: The spectrum σ(A) contains A’s eigenvalues, and the spectral radius ρ(A) is the maximum modulus among them.Nonsingularity is equivalent to excluding 0 from the spectrum, or equivalently to det A ≠ 0.
- Spectral concepts: An eigenvalue is semisimple when its geometric and algebraic multiplicities coincide; a matrix is nondefective when this holds for every eigenvalue.Equivalently, nondefective matrices have diagonal Jordan canonical form, while defective matrices do not.
- Canonical forms: The Jordan canonical form represents every complex square matrix as a direct sum of Jordan blocks and is unique up to block permutation.A Jordan block J_k(λ) has algebraic multiplicity k and geometric multiplicity 1 for λ.
- Basic properties: The spectral radius obeys ρ(A^k) = (ρ(A))^k, so (ρ(A^k))^1/k equals ρ(A) for every natural number k.The spectral radius is also the greatest lower bound of the values of all induced matrix norms.
2 Framework
The framework extends the spectral radius from one matrix to matrix families and connects it to stability and robustness of uncertain discrete-time systems. For bounded families, four spectral-radius characterizations coincide, while noncommutativity and unboundedness impose important limits.
- Stability and robustness: Time-varying perturbations model incomplete modeling, neglected dynamics, or measurement uncertainty, motivating robustness analysis of whether nominal stability persists.The perturbation sequence is not known a priori and may vary arbitrarily within a known uncertainty bound.
- Stability and robustness: Uniform asymptotic stability requires every possible left product of matrices from the family to vanish as its length tends to infinity.This generalizes the single-matrix criterion that the spectral radius is strictly less than one.
- Spectral-radius definitions: The joint spectral radius measures worst-case asymptotic growth across all products formed from a family of matrices.Products of length k are collected in P_k(F), whose normalized maximal norms define the limiting quantity.
- Properties and scope: The joint spectral radius is independent of the chosen matrix norm, but exploiting it remains intrinsically difficult because matrix multiplication is noncommutative.The framework subsequently assumes bounded matrix sets, under which the common spectral-radius characterization is available.
- Equivalence results: The generalized spectral radius can be expressed through the asymptotic limsup of normalized spectral radii of products, including cases where no finite product attains the limiting value.When a finite product attains the value, its powers attain it at every corresponding length; otherwise the supremum is approached only as length tends to infinity.
- Equivalence results: For bounded matrix families, the generalized, joint, common, and mutual spectral radii are equal.The complete spectral radius theorem establishes coincidence of the four characterizations.
6. Block triangular matrices: Given a family of block upper triangular matrices
The joint spectral radius characterizes uniform asymptotic stability and supports several structural reductions, but its general computation and related decision problems are difficult. Important special cases yield exact values or decidability results, while finiteness and algebraic characterizations remain limited.
- Stability: ρ(F) < 1 if and only if the associated system is uniformly asymptotically stable.The criterion applies to bounded families of matrices.
- Product behavior: If products from a bounded family converge, its multiplicative monoid is bounded, but the converse fails in general.
- Special cases: For stochastic families, ρ(F) = 1 because stochastic matrices are closed under multiplication and each has spectral radius 1.
- Special cases: For F = {A,A∗}, the joint spectral radius equals ρ(AA∗)^1/2, which is the largest singular value σ1(A).
- Complexity and decidability: For k,n ≥ 2, the sets of matrix tuples with ρ(x) < 1 and with bounded semigroups are not semi-algebraic.