Joint Spectral Subradius Overview

• Joint spectral subradius is the minimal exponential growth rate of long matrix products, defined via an invariant limit independent of the operator norm.
• It exhibits discontinuity in general but attains continuity under invariant cone structures, which is crucial for stability analysis in control systems.
• Exact computation methods using antinorms on cones enable practical evaluation of the subradius in switched systems and nonnegative matrix families.

Searching arXiv for the cited papers and closely related lower spectral radius results. arxiv_search(query="joint spectral subradius lower spectral radius continuity invariant cone antinorm", max_results=10) The joint spectral subradius, also called the lower spectral radius, of a bounded or compact set of real matrices is the smallest possible exponential growth rate of long products of matrices drawn from that set. For a bounded set $\Sigma\subset\mathbb R^{n\times n}$ and any operator norm $\|\cdot\|$ on $\mathbb R^{n\times n}$, it is defined by

$$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$

and the limit exists and does not depend on the choice of norm. In the literature this quantity is studied as a spectral invariant of matrix semigroups and linear switched systems; it also arises naturally in connection with combinatorics on words, the stability of linear inclusions in control theory, and the study of random Cantor sets (Jungers, 2012, Bochi et al., 2013).

1. Definition and equivalent formulations

Let $\mathcal A$ be a nonempty compact set of real $d\times d$ matrices, or a finite family $\mathcal A=\{A_1,\dots,A_m\}$. The lower spectral radius is given by

$$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$

Existence of the limit follows from submultiplicativity and Fekete’s lemma. For a finite set $\mathcal A$, the equivalent formula

$$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$

shows that the operator norm may be replaced by the spectral radius. In all of these formulations, the quantity is independent of the choice of norm and measures the smallest possible exponential growth rate of long products from the family (Guglielmi et al., 2011).

Several basic inequalities place the joint spectral subradius within a broader hierarchy of joint spectral characteristics. One has

$\|\cdot\|$0

where $\|\cdot\|$1. For switched-system analysis, this hierarchy is important because perturbations or structural assumptions may force the lower spectral radius to coincide with one of these exterior-power quantities.

2. Discontinuity and upper-semicontinuity

Unlike the usual upper joint spectral radius, the joint spectral subradius need not be continuous in general. It is always upper-semi-continuous on the space of compact matrix sets, but without further assumptions it may jump downward under arbitrarily small perturbations.

A standard counterexample is given by

$\|\cdot\|$2

for which $\|\cdot\|$3 in the Hausdorff metric, yet

$\|\cdot\|$4

This exhibits the fact that the map $\|\cdot\|$5 is only upper-semi-continuous in general (Jungers, 2012).

This discontinuity has direct methodological consequences. It implies that naive approximation of a matrix family by nearby families can destroy the limiting decay rate, and it shows that continuity of the lower spectral radius is not a formal consequence of compactness, boundedness, or the definition via long products. A common misconception is that invertibility or small perturbation theory alone should suffice; the known results show that additional geometric structure is required.

3. Invariant cones, embedded pairs, and continuity restoration

A major continuity theorem becomes available when the matrices share an invariant cone structure. A proper cone $\|\cdot\|$6 is a closed, convex, pointed cone with nonempty interior. A matrix $\|\cdot\|$7 is said to be $\|\cdot\|$8-nonnegative if $\|\cdot\|$9. If $\mathbb R^{n\times n}$0 and $\mathbb R^{n\times n}$1 are two proper cones, then $\mathbb R^{n\times n}$2 is embedded in $\mathbb R^{n\times n}$3 if

$\mathbb R^{n\times n}$4

For an embedded pair $\mathbb R^{n\times n}$5, one associates a finite constant $\mathbb R^{n\times n}$6 defined by the requirement that for every line $\mathbb R^{n\times n}$7 that meets $\mathbb R^{n\times n}$8 in a segment $\mathbb R^{n\times n}$9 and meets $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$0 in $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$1 with $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$2,

$$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$3

An embedded pair $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$4 is invariant for a set $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$5 if every $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$6 satisfies $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$7 and $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$8.

Under this hypothesis, continuity is restored: if $$\check\rho(\Sigma)=\lim_{t\to\infty}\min\Big\{\big\|A_{i_1}A_{i_2}\cdots A_{i_t}\big\|^{1/t}:A_{i_j}\in\Sigma\Big\},$$9 is a compact set that leaves invariant an embedded pair of cones $\mathcal A$0, and if $\mathcal A$1 is a sequence of compact matrix sets converging to $\mathcal A$2 in the Hausdorff metric, then

$\mathcal A$3

Equivalently, for every $\mathcal A$4 there exists $\mathcal A$5 such that whenever the Hausdorff distance $\mathcal A$6, one has

$\mathcal A$7

The proof uses an auxiliary lemma stating that if there exists $\mathcal A$8 and $\mathcal A$9 such that $d\times d$0 for all $d\times d$1, then $d\times d$2, together with an extremal-vector theorem for embedded pairs: there exists a nonzero $d\times d$3 such that

$d\times d$4

Any set of strictly positive matrices satisfies the embedded-pair condition, so $d\times d$5 is continuous in a neighborhood of any set of positive matrices. The invariant-cone hypothesis is central: it yields a common “order structure” on products, allows the construction of a positive invariant vector, and recovers a weak Perron–Frobenius setting (Jungers, 2012).

The same framework gives a natural interpretation in switching theory. The joint spectral subradius is a key quantity in worst-case decay for switching systems, so continuity under the embedded-cone assumption means that small perturbations of the matrices lead to small changes in the subradius. A plausible implication is that discretized or simplified families can be used as approximants when the embedded-pair hypothesis is preserved.

4. Domination, continuity criteria, and lower finiteness pathology

For compact sets of invertible matrices, continuity can be characterized in terms of dominated splittings. A compact set $d\times d$6 is called $d\times d$7-dominated, for $d\times d$8, if there exist constants $d\times d$9 and $\mathcal A=\{A_1,\dots,A_m\}$0 such that for every product $\mathcal A=\{A_1,\dots,A_m\}$1 with $\mathcal A=\{A_1,\dots,A_m\}$2 one has

$\mathcal A=\{A_1,\dots,A_m\}$3

where $\mathcal A=\{A_1,\dots,A_m\}$4 are the singular values. Equivalently, there is a continuous invariant multicone in $\mathcal A=\{A_1,\dots,A_m\}$5 separating an expanding $\mathcal A=\{A_1,\dots,A_m\}$6-plane from a contracting $\mathcal A=\{A_1,\dots,A_m\}$7-plane.

Within the class of 1-dominated compact sets, the lower spectral radius is locally Lipschitz. The proof constructs a lower Barabanov function $\mathcal A=\{A_1,\dots,A_m\}$8 on the multicone $\mathcal A=\{A_1,\dots,A_m\}$9, namely a continuous function $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$0 satisfying

$$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$1

and homogeneous of degree one. More generally, if $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$2 denotes the smallest index $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$3 for which $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$4 is $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$5-dominated, then

$$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$6

In particular, $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$7 is continuous at $$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$8 if and only if

$$\underline{\rho}(\mathcal A):=\lim_{n\to\infty}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\} =\inf_{n\ge 1}\inf\{\|A_{i_n}\cdots A_{i_1}\|^{1/n}:A_{i_j}\in\mathcal A\}.$$9

This continuity theory is closely tied to a negative result for finite products. A finite set $\mathcal A$0 has the lower finiteness property if there exists a finite product $\mathcal A$1 attaining the infimum,

$\mathcal A$2

On a certain open set $\mathcal A$3, the set

$\mathcal A$4

is a dense $\mathcal A$5 in $\mathcal A$6, and every $\mathcal A$7 fails to have the lower finiteness property. The explicit $\mathcal A$8 example of Bousch–Mairesse,

$\mathcal A$9

has $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$0 but no product has spectral radius $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$1. These results show that brute-force algorithms that search for a minimizing finite product cannot be expected to succeed generically, even though domination provides a pathway toward continuity and, in some cases, computation (Bochi et al., 2013).

5. Exact computation through antinorms on cones

An exact computational framework for the lower spectral radius is available for cone-preserving families. Let $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$2 be a convex cone that is pointed, closed, and has nonempty interior. A function $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$3 is an antinorm if it is continuous on $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$4, positively homogeneous, concave on $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$5, and not identically zero. An antinorm is monotone if $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$6. It is extremal if

$$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$7

If $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$8 satisfies $$\check\rho(\mathcal A)=\inf_k\min_{B\in\mathcal A^k}\rho(B)^{1/k}\le \inf_k\min_{B\in\mathcal A^k}\|B\|^{1/k}$$9, then $\|\cdot\|$00. If $\|\cdot\|$01 shares a common invariant cone $\|\cdot\|$02, then there exists a monotone extremal antinorm on $\|\cdot\|$03.

This antinorm theory underlies Algorithm (L), an exact procedure for the lower spectral radius of a nonnegative family $\|\cdot\|$04. The input is a nonnegative family and an integer $\|\cdot\|$05 specifying the maximum product length to search. The algorithm chooses a candidate product $\|\cdot\|$06 minimizing $\|\cdot\|$07, normalizes the family by $\|\cdot\|$08, computes cyclic permutations of $\|\cdot\|$09, and iteratively constructs an “infinite polytope”

$\|\cdot\|$10

At each step it solves the LP

$\|\cdot\|$11

to decide whether a new image $\|\cdot\|$12 lies in the current polytope. If no new vertices appear, the polytope is invariant, the associated Minkowski-type functional is an extremal antinorm, and the algorithm returns the exact value of $\|\cdot\|$13. If the algorithm does not terminate, it produces lower and upper bounds that converge to the exact value.

Finite termination depends on additional structure. A product $\|\cdot\|$14 is under-dominant if there exists $\|\cdot\|$15 such that every product in the normalized family that is not a power or cyclic permutation of $\|\cdot\|$16 has spectral radius $\|\cdot\|$17. A nonnegative family is eventually positive if some power of each product is strictly positive, equivalently if there exists a subcone $\|\cdot\|$18 invariant under all $\|\cdot\|$19. If $\|\cdot\|$20 is eventually positive, then Algorithm (L) terminates in finite steps if and only if the chosen candidate $\|\cdot\|$21 is under-dominant. In numerical experiments, the algorithm for the lower spectral radius is reported to work fast for nonnegative matrices, with random tests in dimensions up to $\|\cdot\|$22 producing exact values within a few seconds to minutes, typically with $\|\cdot\|$23–$\|\cdot\|$24 iterations and $\|\cdot\|$25 vertices (Guglielmi et al., 2011).

6. Switched systems, stabilizability, and singular-matrix regimes

In discrete-time switched systems, the lower spectral radius appears alongside the upper joint spectral radius and the stabilizability radius. For a finite set $\|\cdot\|$26 of cardinality $\|\cdot\|$27,

$\|\cdot\|$28

and

$\|\cdot\|$29

It is immediate from the definitions that

$\|\cdot\|$30

A general lower bound strengthens this relation: $\|\cdot\|$31

Equality $\|\cdot\|$32 holds in several singular settings. A set $\|\cdot\|$33 is called irreducible if no proper subset $\|\cdot\|$34 has the same stabilizability radius, meaning

$\|\cdot\|$35

If $\|\cdot\|$36 is irreducible and contains at least one singular matrix whose image is one-dimensional, then

$\|\cdot\|$37

In particular, any irreducible $\|\cdot\|$38 with at least one singular matrix satisfies $\|\cdot\|$39.

A detailed $\|\cdot\|$40 example is the singular-plus-rotation system

$\|\cdot\|$41

Here $\|\cdot\|$42 has one nonzero eigenvalue $\|\cdot\|$43 and one zero eigenvalue, and if $\|\cdot\|$44 denotes the acute angle between the two real eigenvectors of $\|\cdot\|$45, then

$\|\cdot\|$46

Three cases are distinguished. If $\|\cdot\|$47 for some $\|\cdot\|$48, then $\|\cdot\|$49. If $\|\cdot\|$50 but $\|\cdot\|$51 for all $\|\cdot\|$52, then

$\|\cdot\|$53

If $\|\cdot\|$54 and $\|\cdot\|$55 for all $\|\cdot\|$56, then continued-fraction convergents of $\|\cdot\|$57 yield a sequence $\|\cdot\|$58 such that

$\|\cdot\|$59

The parameter sets for which the stabilizability radius takes a prescribed value can be extremely thin. For fixed $\|\cdot\|$60 and constant $\|\cdot\|$61, each set

$\|\cdot\|$62

has Hausdorff dimension zero; in the irrational case the subset $\|\cdot\|$63 also has Hausdorff dimension zero, and when $\|\cdot\|$64, the zero-radius set $\|\cdot\|$65 lies inside the classical set of Liouville numbers. These results place the joint spectral subradius at the center of a precise interface between algebraic structure, Diophantine approximation, and switched-system stabilizability (Dettmann et al., 22 Sep 2025).