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We suppose without loss of generality that ρˆ (Σ ) = 1 (nondefectivity is clearly conserved under scalar multiplication). Let us define V = {x ∈ Rn | sup |Ax| < ∞}. A∈Σ ∗ By construction, V is a linear subspace, invariant under the matrices in Σ . Since Σ is irreducible, we have V = Rn , or V = {0}. If V = {0}, then for each vector x ∈ Rn , there exists a product A ∈ Σ t such that |Ax| ≥ 2|x|. We claim that this length t is bounded uniformly from above by a constant T over all x. Indeed, if it is not the case, we can define an increasing sequence {tk }, and a sequence xk of norm 1 such that for all A ∈ Σ ≤tk , |Axk | < 2.

John’s ellipsoid theorem” [56]) Let K ∈ Rn be a compact convex set with nonempty interior. Then there is an ellipsoid E with center c such that the inclusions E ⊂ K ⊂ n(E − c) + c hold. √ If K is symmetric about the origin (K = −K), the constant n can be changed into n. 11. [2, 15] For an arbitrary set of m matrices Σ ⊂ Rn×n , the best ellipsoidal norm approximation ρ ∗ of its joint spectral radius ρ satisfies 1 √ ρ ∗ ≤ ρ ≤ ρ ∗, n 1 √ ρ ∗ ≤ ρ ≤ ρ ∗. 3 Methods of Computation 41 Proof. The first part is a simple application of John’s ellipsoid theorem.

6 we formulate several open problems on possible generalizations of those results to arbitrary matrices. 2 Auxiliary Facts and Notations For two nonnegative functions f (t), g(t) we use the standard notation f (t) = O(g(t)), which means that there is a positive constant C such that f (t) ≤ Cg(t) for all t. The functions f and g are said to be asymptotically equivalent, which we denote f (t) g(t) if f (t) = O(g(t)) and g(t) = O( f (t)). We shall consider each nonnegative n × n matrix A as the adjacency matrix of a directed weighted graph G(A).

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