Induced-Norm Contraction Factor

• Induced-norm contraction factor is a measure that quantifies the minimum contraction rate achievable through optimal absolute vector norm selection of a matrix map.
• It connects to joint spectral radius and matrix measures, serving as a key parameter in stability analysis, robust control, and invariant set approximations.
• Norm shaping techniques, such as diagonal rescaling and Lyapunov norm optimization, effectively reduce conservatism in convergence rates for dynamic systems.

The induced-norm contraction factor quantifies the minimal rate at which a linear or nonlinear map contracts distances, as measured in vector norms whose induced matrix norms satisfy natural invariance or monotonicity properties. This concept pervades stability analysis, robust control, Lyapunov theory, and invariant set approximations, appearing under various names: minimal induced norm, contraction rate, matrix measure, logarithmic norm, and joint spectral radius. The induced-norm contraction factor is tightly linked to optimal norm selection for minimizing conservativeness in convergence rates and for certifying stability or contractivity properties of dynamical systems and matrix operators.

1. Definitions and Fundamental Formulation

For a given matrix $A\in\mathbb{F}^{n\times n}$ (where $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$) and an absolute vector norm $v:\mathbb{F}^n\to[0,\infty)$ (i.e., $v(x)=v(|x|)$), the induced matrix norm is defined by

$$\|A\|_v := \sup_{x\ne 0} \frac{v(Ax)}{v(x)} = \max_{v(x)=1} v(Ax).$$

The induced-norm contraction factor $F(A)$ is the infimum of the induced norms over all absolute norms,

$F(A) := \inf_{v \text{ absolute}} \|A\|_v.$

The contraction factor thus measures how much the action of $A$ can be made to contract vectors, with the "best-case" contraction arising from optimizing over all admissible norms (Friedland, 2021).

For dynamical systems, the contraction rate $c$ associated with a norm $\|\cdot\|$ for the system $\dot{x} = f(x)$ is tied to the matrix measure (logarithmic norm) $\mu(A)$,

$$\mu(A) := \lim_{h\searrow 0} \frac{\|I + hA\| - 1}{h}.$$

The largest $c$ such that

$\|\phi(t,x) - \phi(t,y)\| \le e^{-ct}\|x-y\|$

for all initial pairs $(x,y)$ and $t\ge 0$ is the norm-dependent contraction factor for the flow $f$ (Coogan et al., 2013).

2. Main Characterizations and Theorems

The minimal induced-norm contraction factor $F(A)$ admits two fundamental characterizations:

• Infimum Over Norms: $F(A)$, as above, over all absolute norms.
• Joint Spectral or "limsup" Characterization: Define the set of diagonal unimodular matrices $$\mathcal{D} := \{\mathrm{diag}(d_1,\dots,d_n) : |d_i|=1\}$$. Then

$$F(A) = \limsup_{k\to\infty} \max_{D_1,\dots,D_k\in \mathcal{D},\, x:\|x\|_2=1} \Bigl(\|D_1 A D_2 A\cdots D_k A\,x\|_2\Bigr)^{1/k}.$$

This "limsup" formula demonstrates that $F(A)$ is the joint spectral radius of the set $\{ D A : D\in\mathcal{D} \}$ (Friedland, 2021).

There are universal bounds:

$\rho(A) \le F(A) \le \rho(|A|),$

where $\rho(A)$ is the spectral radius of $A$ and $|A|$ is the matrix of elementwise moduli. Equality $F(A)=\rho(|A|)$ holds precisely when $A$ is sign-equivalent to $|A|$, i.e., $A = D_1 |A| D_2$ for some $D_1,D_2\in\mathcal{D}$.

For finite-horizon truncations of sets such as the minimal robust positive invariant (mRPI) set, the contraction factor determines the rate at which the truncation error decays geometrically (Sun, 23 Nov 2025).

3. Computation via Matrix Measures and Optimization

The computation or estimation of induced-norm contraction factors depends on the norm:

• $\ell_2$ (Euclidean): $\|A\|_2 = \sigma_{\max}(A)$ (maximal singular value); matrix measure (log norm) $\mu_2(A) = \lambda_{\max}((A+A^T)/2)$.
• $\ell_1$ and $\ell_\infty$: $\|A\|_1$ is maximal column sum, $\|A\|_\infty$ is maximal row sum. $\mu_1(A) = \max_j(a_{jj} + \sum_{i\ne j}|a_{ij}|)$, $$\mu_\infty(A) = \max_i(a_{ii}+\sum_{j\ne i}|a_{ij}|)$$.
• Weighted Euclidean ($\|x\|_P = \sqrt{x^T P x}$): $\|A\|_P = \sigma_{\max}(P^{1/2} A P^{-1/2})$, minimized by solving a Lyapunov inequality $A^T P A \preceq \gamma^2 P$ via LMI.
• Polyhedral Norms: Computation reduces to convex or linear programs, e.g., for the max norm, $$\mu_\infty(A) = \max_i(a_{ii}+\sum_{j\ne i}|a_{ij}|)$$; for general norms, the induced matrix measure can be formulated via regular pairings and convex programming (Proskurnikov et al., 30 Aug 2024).

In control applications, one minimizes $\gamma = \|A\|$ under norm shaping to accelerate convergence:

• Step 1: Compute the contraction in the Euclidean norm.
• Step 2: Diagonal rescaling via a weighted norm (coordinate search, SDP) to further lower $\gamma$.
• Step 3: Apply full Lyapunov norm optimization for the best (quadratic) contraction rate—$\gamma$ typically approaches $\rho(A)$ from above as the feasible region is enlarged (Sun, 23 Nov 2025).

4. Role in Stability, Invariance, and Contraction Analysis

The induced-norm contraction factor is central to discrete- and continuous-time stability, reachability, and robustness results:

• Exponential Contraction: For $\dot{x}=f(x)$, if the supremum of the matrix measure $\mu(A)$ over the relevant state set is $\le -c$, then the system is contractive at rate $c$ in the selected norm (Coogan et al., 2013, Harapanahalli et al., 18 Nov 2024).
• Bounded Error in Truncated Invariant Sets: For Schur stable $A$, the error in the truncated mRPI set is explicitly bounded as

$$d_H(E_N, E_\infty) \le r_W \, \gamma^{N} / (1-\gamma),$$

where $r_W = \max_{w\in W} \|w\|$ and $\gamma := \|A\|$ is the induced-norm contraction factor. A judicious norm reduces the required $N$ for a target error by decreasing $\gamma$ (Sun, 23 Nov 2025).

• Lyapunov Functions: The contraction factor underlies construction of Lyapunov functions for certifying global exponential stability via distance or velocity-norm decay (Coogan et al., 2013).
• Contraction to Trajectories: In more advanced settings, e.g., for contraction to a known trajectory, tighter induced-norm contraction bounds arise from the analysis of mixed Jacobians and associated LDIs (linear differential inclusions), enabling less conservative certificates—especially pronounced in the $\ell_1$ norm or when interval overapproximations are employed (Harapanahalli et al., 18 Nov 2024).

5. Computation and Algorithmic Approaches

Norm Type	Closed-Form Formula/Optimization	Sharpness/Role
$\ell_2$	$\sigma_{\max}(A)$ (SVD), $\mu_2$ via eigenvalue	No design freedom; baseline
$\ell_1/\ell_\infty$	Max column/row sum; explicit matrix measures	Often conservative; baseline
Weighted Euclidean	SDP minimizing $\|A\|_P$ or $\mu_P(A)$	Best among quadratics
Polyhedral norms	LP/convex programs (via pairings)	Non-Euclidean, geometry-exploiting

The contraction factor is minimized by norm shaping, for which:

• Diagonal scalings yield rapid (10–50%) reductions in $\gamma$ at modest computational cost.
• Full Lyapunov norm optimization, solved via LMIs, realizes the minimal contraction in quadratic norms.
• In high dimensions, random stable $A$ can exhibit large improvements via norm shaping, justifying non-Euclidean and problem-adapted norms (Sun, 23 Nov 2025, Proskurnikov et al., 30 Aug 2024).

For joint spectral radius computations (needed for exact $F(A)$ for arbitrary $A$), one employs algorithms such as Gripenberg's method or ellipsoid technique, especially for finite matrix sets arising via sign transformations (Friedland, 2021).

6. Special Cases, Extensions, and Theoretical Significance

When $A$ is sign-equivalent to a nonnegative matrix $B$, the minimal induced-norm contraction factor $F(A)$ coincides with the spectral radius of $B$, and hence of $|A|$. In this setting, absolute vector norms (e.g., weighted $\ell_1$ or $\ell_\infty$) attain the theoretical lower bound, as guaranteed by the Perron-Frobenius theory. If $|A|$ is reducible, the minimal contraction can be approached via generalizations of the Collatz–Wielandt formula, with convergence to the spectral radius as $\varepsilon\to 0$ (Friedland, 2021).

Regular pairings associated with general norms unify and extend contraction theory beyond the Euclidean world. The existence of such pairings ensures that the curve-norm derivative formula and Lumer's inequality hold, linking contraction rates, Lyapunov derivatives, and induced logarithmic norms. For polyhedral norms, the contraction factor can be exactly computed as the solution of finitely many linear inequalities, facilitating efficient analysis in structured or sparse systems (Proskurnikov et al., 30 Aug 2024).

Interval overapproximations and mixed-Jacobian analysis can yield strictly less conservative contraction certificates, especially for systems with structure or when considering trajectories rather than invariant sets. In particular, the mixed Jacobian approach provides a provably tighter log norm in the $\ell_1$ norm than traditional Jacobian-based interval approaches (Harapanahalli et al., 18 Nov 2024).

7. Applications and Impact Across Areas

• Robust Invariant Set Computation: Direct use of the induced-norm contraction factor yields analytic bounds on the truncation error for mRPI sets, enabling set-based model predictive control (MPC) synthesis and constraint tightening without heavy iterative computations (Sun, 23 Nov 2025).
• Stability and Reachability Analysis: Contraction factors, particularly the tightest ones obtainable by norm shaping, underpin system-theoretic certificates of global exponential stability and estimations of reachable sets for ODEs and switched/differential inclusion systems (Coogan et al., 2013, Harapanahalli et al., 18 Nov 2024).
• Norm-Design in Control and Optimization: The induced-norm contraction factor is a natural design parameter for optimizing convergence, accelerating algorithmic performance, and certifying properties under minimal conservativeness. The theory connects deeply with Lyapunov analysis, operator theory, and spectral graph methods.

The concept thus acts as a cornerstone in the quantitative analysis and certification of contraction, stability, and robustness across matrix and operator theory, nonlinear dynamics, and set-based control (Friedland, 2021, Sun, 23 Nov 2025, Coogan et al., 2013, Harapanahalli et al., 18 Nov 2024, Proskurnikov et al., 30 Aug 2024).