Hadamard functions preserving nonnegative H-matrices Original Research Article

For k nonnegative n × n matrices Al = (alij) and a function f :imagek+ → image., consider the matrix imageC = f(A1,…,Ak) = (cij), where imageCij = f(A1ij,…,Akij), i, J = 1, …n. Denote by ρ(A) the spectral radius of a nonnegative square matrix A, and by σ(A) the minimal real eigenvalue of its comparison matrix M(A) = 2 diag(aii) − A. It is known that the function imagef(X1,…, Xk = cxX11,cdots, three dots, centered,XXkk), where xi set membership, variant image.. Σki=1 xi ≥ 1 and c > 0, satisfies the inequalities imagep(f(A1,…,Ak)) ≤ f(p(A1), …p(Ak)), as well as the inequalities imageσ(f(A1,…, Ak)) ≤ f(σA1), …, σ(Ak)), whenever Ai are nonnegative H-matrices, i.e. σ(Ai) ≥ 0. The last inequality implies that the above function f maps the set of nonnegative H-matrices into itself. In this note it is proven that these are the only continuous functions with this property.