Barabanov Norm: Extremal Norm in Matrix Analysis

• Barabanov norm is an extremal norm for a family of linear operators that ensures the worst-case one-step expansion equals the joint spectral radius in every direction.
• It plays a pivotal role in joint spectral radius theory, switched-system stability, and invariant convex geometry, and extends to both discrete and continuous time systems.
• Constructive methods like max-relaxation and invariant polytope algorithms provide explicit formulations and numerical approximations, enhancing analysis of asymptotic behavior.

A Barabanov norm is an extremal norm adapted to the maximal asymptotic growth of a family of linear operators. In the classical discrete-time setting, for a bounded irreducible family $\mathcal{A}$ of matrices with joint spectral radius $\rho(\mathcal{A})$, it is a norm $\|\cdot\|_B$ such that

$$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$

so the worst one-step expansion is exactly the joint spectral radius in every direction. This places Barabanov norms at the intersection of joint spectral radius theory, switched-system stability, extremal trajectory analysis, and invariant convex geometry. In later developments, the notion has been extended or specialized to continuous-time switching systems, max algebra, constrained switching classes, and fiber-bunched cocycles, while preserving the same structural role: a norm or Finsler norm that realizes the maximal asymptotic growth rate in a pointwise extremal manner (Morris, 2011, Guglielmi et al., 2017, Bochi et al., 2018).

1. Classical definition and relation to the joint spectral radius

For a bounded family $\mathcal{A}\subset M_d(\mathbb{K})$, the joint spectral radius is the maximal exponential growth rate of long products of matrices drawn from $\mathcal{A}$. In the notation of Morris, for a bounded set $\mathcal{A}$,

$$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$

and this limit is independent of the chosen norm (Morris, 2011). In the notation of Kozyakin for a finite family $\mathcal{A}=\{A_1,\dots,A_r\}$,

$$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$

with $\rho(\mathcal{A})$0 defined through maximal norms of products; for finite $\rho(\mathcal{A})$1, this coincides with the generalized spectral radius $\rho(\mathcal{A})$2 (Kozyakin, 2010).

Under irreducibility, Barabanov’s theorem yields a norm satisfying the Bellman-type equation

$\rho(\mathcal{A})$3

or equivalently,

$\rho(\mathcal{A})$4

This norm is the Barabanov norm (Morris, 2011, Kozyakin, 2010). It is stronger than a general extremal norm: an extremal norm only requires $\rho(\mathcal{A})$5 for all $\rho(\mathcal{A})$6, whereas a Barabanov norm requires equality after maximizing over the family for every vector. Consequently, it not only bounds the semigroup sharply but also encodes maximizing directions and maximizing switching laws (Morris, 2011, Kozyakin, 2 Sep 2025).

An equivalent geometric formulation uses invariant convex bodies. If $\rho(\mathcal{A})$7 is the unit ball of a Barabanov norm and all matrices are invertible, then

$\rho(\mathcal{A})$8

Dual formulations replace $\rho(\mathcal{A})$9 by an invariant body for the transpose family or by a Dranishnikov–Konyagin body satisfying

$\|\cdot\|_B$0

The duality between these two formulations is mediated by polarity, and the corresponding Minkowski functional of a centrally symmetric invariant body yields an extremal norm (Kozyakin, 2 Sep 2025, Protasov, 2021).

2. Existence, irreducibility, and continuous-time analogues

The standard existence theorem requires irreducibility. In the discrete-time setting, irreducibility means the family has no nontrivial common invariant subspace; under compactness and irreducibility, Barabanov norms exist (Morris, 2011, Kozyakin, 2010). This existence statement is fundamental but nonconstructive in its original form.

In continuous time, the analogous object is defined for a compact irreducible set $\|\cdot\|_B$1 governing the switching system

$\|\cdot\|_B$2

The asymptotic growth rate is then the top Lyapunov exponent, denoted $\|\cdot\|_B$3 in the planar switching literature and $\|\cdot\|_B$4 or $\|\cdot\|_B$5 in other continuous-time treatments. After shifting by $\|\cdot\|_B$6 or $\|\cdot\|_B$7, one often reduces to the normalized case where the maximal exponent is $\|\cdot\|_B$8. In that normalization, a continuous-time Barabanov norm is a norm $\|\cdot\|_B$9 such that $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$0 is nonincreasing along every trajectory, and from every initial state there exists an extremal trajectory along which $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$1 is constant (Protasov et al., 2024, Morris, 2023, Chitour et al., 2014).

This formulation generalizes the discrete-time saturation property. In the notation of Chitour–Mason–Sigalotti, when $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$2, a Barabanov norm $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$3 satisfies

$$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$4

for every trajectory, and for every $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$5 there exists a trajectory $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$6 with $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$7 such that

$$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$8

The same pattern appears in Morris’s continuous-time formulation, where extremal trajectories keep $$\max_{A\in\mathcal{A}} \|Ax\|_B = \rho(\mathcal{A})\|x\|_B \quad \forall x,$$9 constant (Chitour et al., 2014, Morris, 2023).

The notion also extends beyond ordinary finite switching sets. For fiber-bunched cocycles and vector bundle automorphisms over a hyperbolic homeomorphism, an extremal Finsler norm satisfies

$\mathcal{A}\subset M_d(\mathbb{K})$0

and in the one-step symbolic case this reduces to a classical Barabanov norm on $\mathcal{A}\subset M_d(\mathbb{K})$1 (Bochi et al., 2018). For constrained switching classes not closed under concatenation, a generalized Barabanov theory is recovered by passing to a concatenable subfamily and constructing a quasi-Barabanov semigroup; this yields extremal norms and quasi-extremal trajectories for dwell-time and related constraints (Chitour et al., 2015).

3. Geometric structure: invariant bodies, extremal trajectories, and differentiability

The unit ball of a Barabanov norm is a centrally symmetric convex body whose boundary organizes extremal dynamics. In discrete time, if $\mathcal{A}\subset M_d(\mathbb{K})$2 is a Barabanov norm, then the unit ball $\mathcal{A}\subset M_d(\mathbb{K})$3 and its polar $\mathcal{A}\subset M_d(\mathbb{K})$4 satisfy dual invariance relations, and the classification of invariant bodies for the transpose family becomes a classification of Barabanov norms (Protasov, 2021, Kozyakin, 2 Sep 2025).

In continuous time, the Barabanov sphere $\mathcal{A}\subset M_d(\mathbb{K})$5 is invariant under extremal trajectories. Extremality can be characterized by a maximum principle. For a Barabanov norm $\mathcal{A}\subset M_d(\mathbb{K})$6 and dual norm $\mathcal{A}\subset M_d(\mathbb{K})$7, if $\mathcal{A}\subset M_d(\mathbb{K})$8 is extremal, then there exists an adjoint trajectory $\mathcal{A}\subset M_d(\mathbb{K})$9 such that

$\mathcal{A}$0

and

$\mathcal{A}$1

Thus extremal trajectories satisfy a Pontryagin-type condition on the support hyperplanes to the Barabanov sphere (Chitour et al., 2014).

This geometry is particularly explicit in the planar continuous-time case. For $\mathcal{A}$2, a norm $\mathcal{A}$3 is Barabanov if, for every point $\mathcal{A}$4 on the unit sphere $\mathcal{A}$5, each vector $\mathcal{A}$6 is either tangent to $\mathcal{A}$7 or directed inside the unit ball, and the set of tangent vectors is nonempty. When there is no real dominance in $\mathcal{A}$8, the unit sphere is exactly one period of a periodic leading trajectory and is $\mathcal{A}$9; under complex dominance it is an ellipse, corresponding to a quadratic norm $\mathcal{A}$0 (Protasov et al., 2024).

More generally, Barabanov norms need not be Riemannian. In the cocycle setting, Appendix B.2 of Bochi–Garibaldi gives a two-dimensional one-step cocycle with

$\mathcal{A}$1

for which the max norm is extremal but no Riemannian extremal norm exists (Bochi et al., 2018). This aligns with the broader observation that Barabanov norms are typically Finsler, polyhedral, or piecewise-quadratic rather than Euclidean.

4. Uniqueness, multiplicity, strict convexity, and sensitivity

Uniqueness of a Barabanov norm always means uniqueness up to multiplication by a positive scalar (Morris, 2011, Morris, 2023). The theory exhibits both strong uniqueness mechanisms and dramatic nonuniqueness.

A sufficient condition based on the asymptotic limit semigroup was given by Morris. If the limit semigroup $\mathcal{A}$2 has the transitivity property that for every pair of nonzero vectors $\mathcal{A}$3 there exist $\mathcal{A}$4 and $\mathcal{A}$5 such that

$\mathcal{A}$6

then $\mathcal{A}$7 has a unique Barabanov norm (Morris, 2011). Concrete instances occur when the limit semigroup contains $\mathcal{A}$8 or $\mathcal{A}$9, or a dense subgroup of rotations. In two-dimensional examples with irrational rotations, uniqueness follows; in the corresponding rational-rotation case, there are uncountably many Barabanov norms (Morris, 2011).

At the opposite extreme, the uniqueness property can be highly sensitive to perturbation. Morris proved the existence of a nonempty open set $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$0 such that both the pairs with a unique Barabanov norm and the pairs without a unique Barabanov norm are dense in $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$1 (Morris, 2011). Thus uniqueness is neither generally open nor robust in full generality.

Later work identified broad generic regimes of uniqueness. Protasov’s classification shows that if a discrete-time family has a unique dominant product with a unique simple leading eigenvalue, then the invariant body is unique; if the eigenvalue is real, the Barabanov norm is piecewise-linear, and if it is complex with irrational argument mod $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$2, the Barabanov norm is piecewise-quadratic and unique (Protasov, 2021). In the rational-angle complex case or when there are several dominant products, infinitely many invariant bodies and hence infinitely many Barabanov norms appear (Protasov, 2021).

Strict convexity is a separate issue from uniqueness. In continuous time, Chitour–Mason–Sigalotti proved that in dimension $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$3, if $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$4 is compact, convex, irreducible, all matrices are nonsingular, and $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$5, then Barabanov balls are strictly convex (Chitour et al., 2014). They also showed that if $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$6 is a $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$7 convex compact domain in matrix space, then strict convexity holds in any dimension (Chitour et al., 2014).

However, uniqueness does not imply strict convexity. Morris constructed a four-dimensional continuous-time irreducible switching system with three switching states, every matrix Hurwitz, a unique Barabanov norm up to scalar multiplication, and a Barabanov norm that is not strictly convex (Morris, 2023). This resolves negatively the question of whether uniqueness or the absence of zero eigenvalues forces strict convexity.

5. Constructive theories and explicit formulas

A major development in the theory has been the transition from nonconstructive existence to explicit construction.

Kozyakin introduced a max-relaxation iteration that starts from an arbitrary norm $$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$8 and updates via

$$\varrho(\mathcal{A})=\lim_{n \to \infty} \sup\left\{\left\|A_{i_n} \cdots A_{i_1}\right\|^{\frac{1}{n}} : A_i \in \mathcal{A}\right\},$$9

where $\mathcal{A}=\{A_1,\dots,A_r\}$0 is built from current lower and upper estimates $\mathcal{A}=\{A_1,\dots,A_r\}$1 of the joint spectral radius. For irreducible finite families, $\mathcal{A}=\{A_1,\dots,A_r\}$2 is nondecreasing, $\mathcal{A}=\{A_1,\dots,A_r\}$3 is nonincreasing, both converge to $\mathcal{A}=\{A_1,\dots,A_r\}$4, and the renormalized norms converge uniformly on bounded sets to a Barabanov norm (Kozyakin, 2010). This yields both a numerical construction of the norm and two-sided a posteriori error bounds for the joint spectral radius.

A later reformulation exploits the duality with Dranishnikov–Konyagin bodies. Starting from a centrally symmetric convex body $\mathcal{A}=\{A_1,\dots,A_r\}$5, the convex-hull-relaxation algorithm iterates

$\mathcal{A}=\{A_1,\dots,A_r\}$6

with calibration at each step. The resulting bodies converge in Hausdorff metric to a Dranishnikov–Konyagin body $\mathcal{A}=\{A_1,\dots,A_r\}$7, and the corresponding polar yields a Barabanov norm for the transpose family (Kozyakin, 2 Sep 2025). This avoids matrix inverses and is especially convenient when singular matrices are present.

For generic discrete-time systems, the invariant polytope algorithm is more powerful. When a family has a unique dominant product with unique simple leading eigenvalue, the algorithm terminates in finite time and constructs the invariant body exactly. In the real leading-eigenvalue case, the resulting Barabanov norm is polyhedral; in the complex irrational-angle case, it is the maximum of finitely many quadratic expressions of the form

$\mathcal{A}=\{A_1,\dots,A_r\}$8

This gives explicit finite-dimensional descriptions of the norm in the vast majority of tested examples (Protasov, 2021).

The max-algebraic case is even more explicit. For a compact family $\mathcal{A}=\{A_1,\dots,A_r\}$9 with irreducible semigroup in the cone-theoretic sense, the max-algebraic joint spectral radius equals the spectral radius of the supremum matrix

$$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$0

In that setting, a monotone Barabanov norm always exists and has the form

$$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$1

where $$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$2 is a left max-eigenvector of $$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$3 satisfying

$$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$4

This explicit weighted $$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$5-type formula is one of the rare cases where the Barabanov norm is available in closed form (Guglielmi et al., 2017).

6. Applications, asymptotic dynamics, and generalizations

Barabanov norms are used to characterize worst-case trajectories. In the classical discrete-time setting, iterating the defining identity shows that for every $$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$6 and every $$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$7,

$$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$8

so there exist products realizing maximal growth at every horizon (Morris, 2011). Under finite-dominant-product assumptions, Protasov showed that the only non-decaying trajectories are those generated by eventually periodic switching laws whose period is a dominant product (Protasov, 2021).

In continuous time, extremal trajectories induced by Barabanov norms organize the asymptotic dynamics on the unit sphere. In dimension $$\hat{\rho}(\mathcal{A}) := \limsup_{n\to\infty} \left( \hat{\rho}_n(\mathcal{A}) \right)^{1/n},$$9, under Condition G and nonsingularity assumptions, every extremal trajectory tends to a periodic trajectory; this yields a Poincaré–Bendixson theorem for extremal solutions of linear switched systems (Chitour et al., 2014). In the planar case, the entire stability problem and the problem of constructing Barabanov norms can be resolved explicitly for every compact control set of $\rho(\mathcal{A})$00 matrices; when there is no real dominance, the norm is unique and $\rho(\mathcal{A})$01, while real dominance can produce infinitely many norms, including non-smooth ones (Protasov et al., 2024).

Barabanov-type constructions also underpin regularity results for spectral quantities. In max algebra, the explicit monotone Barabanov norm is used to prove local Lipschitz continuity of the max-algebraic joint spectral radius on the space of compact irreducible sets with respect to the Hausdorff metric (Guglielmi et al., 2017). In the cocycle setting, extremal norms for strongly bunched irreducible automorphisms imply local Lipschitz continuity of the maximal Lyapunov exponent in the $\rho(\mathcal{A})$02-topology on the set of spannable automorphisms (Bochi et al., 2018).

Beyond arbitrary switching, generalized Barabanov theory has been adapted to constrained switching laws. For families not closed under concatenation, such as dwell-time or persistent-excitation classes, one works with a concatenable subfamily and constructs a quasi-Barabanov semigroup. Under assumptions A0–A3, every nonzero initial condition admits a quasi-extremal trajectory satisfying

$\rho(\mathcal{A})$03

This framework is then used to characterize finiteness of the $\rho(\mathcal{A})$04-gain of switched control systems via the generalized spectral radius of a minimal realization (Chitour et al., 2015).

For fiber-bunched cocycles, the extremal norm becomes a noncommutative analogue of a subaction in ergodic optimization. Bochi and Garibaldi showed that a strongly bunched irreducible automorphism over a transitive hyperbolic homeomorphism admits an extremal norm; in the symbolic one-step case, this reduces to a classical Barabanov norm, and over subshifts one obtains Barabanov-like calibration along unstable sets (Bochi et al., 2018).

Barabanov norms therefore remain a central object across several branches of spectral and dynamical systems theory. They provide exact extremal Lyapunov functions, encode maximizing products or trajectories, supply geometric invariants such as invariant convex bodies, and support continuity, stability, and classification results in both classical and nonclassical settings (Guglielmi et al., 2017, Bochi et al., 2018, Protasov, 2021).