The generalized spectral radius is a generalization of the spectral radius ρ ( L ) of a single linear map L, i.e. the maximum of the moduli of the eigenvalues of L. For any set L of linear maps and any norm, the joint spectral radius of L is ρ ̂ = ρ ̂ ( L ) : = lim sup k → ∞ ρ ̂ k 1 / k where ρ ̂ k : = sup σ ∈ Ω k ∥ L σ ∥.
(2) For an IFS consisting of a single linear map L, the generalized spectral radius is the ordinary spectral radius of L, the maximum of the moduli of the eigenvalues of L. (3) For any real α > 0 we have ρ ( α L ) = α ρ ( L ) and ρ ̂ ( α L ) = α ρ ̂ ( L ). (4) If L is bounded, then the joint spectral radius equals the generalized spectral radius.
For an IFS consisting of a single linear map L, the generalized spectral radius is the ordinary spectral radius of L, the maximum of the moduli of the eigenvalues of L. For any real α > 0 we have ρ ( α L ) = α ρ ( L ) and ρ ̂ ( α L ) = α ρ ̂ ( L ).
The generalized spectral radius of L is ρ = ρ ( L ) : = lim sup k → ∞ ρ k 1 / k where ρ k : = sup σ ∈ Ω k ρ ( L σ ).
The following are well known properties of the joint and generalized spectral radii.
The |. | 1 a, q -surface content of the whole generalized ellipsoid Ea,p(R) of |.|a,p-radius R is Oa,p,q(Ea,p(R))=a1…a n ωn,pRn−1.
If L is bounded, then the joint spectral radius equals the generalized spectral radius.
What follows is the definition of the joint spectral radius and the generalized spectral radius of L. Let Ω k be the set of all words σ 1 σ 2 ⋯ σ k, of length k, where σ i ∈ I, 1 ≤ i ≤ k.
The size of the nodes depends on their nature: minor metabolites are smaller than the other ones; a radius of a generalized metabolite/reaction is calculated as a sum of radiuses of the elements that it groups; compartment sizes are defined by the layouts of the elements inside them, so that the compartments are represented as minimal rectangles containing all the corresponding elements.
