10.1016/S0024-3795(97)10094-5
Elsner, Ludwig
Ludwig
Elsner
Hershkowitz, Daniel
Daniel
Hershkowitz
Hadamard functions preserving nonnegative H-matrices
ELSEVIER SCIENCE INC
1998
2010-04-28T13:20:31Z
2019-07-08T13:58:30Z
journal_article
https://pub.uni-bielefeld.de/record/1625037
https://pub.uni-bielefeld.de/record/1625037.json
For k nonnegative n x n matrices A(l) = (a(ij)(l)) and a function f : R<INF>+</INF><SUP>k</SUP> --> R<INF>+ </INF>consider the matrix C = f(A(l),...,A(k)) = (C-ij), where c(ij) = f(a(ij)(l),...,a(ij)(k)), i,j=l,...,n. Denote by rho(A) the spectral radius of a nonnegative square matrix A, and by sigma(A) the minimal real eigenvalue of its comparison matrix M(A) = 2 diag(a(ii)) -A. It is known that the function f(x(1),...,x(k)) = cx(1)(alpha 1)...x(k)(alpha k), where alpha(i) is an element of R+, Sigma(i=l)(k) alpha(i) greater than or equal to 1 and c > 0, satisfies the inequalities rho(f(A(1),...,A(k))) less than or equal to f(rho(A(1)),...,rho(A(k))), as well as the in equalities sigma(f(A(1),...,A(k))) greater than or equal to f(sigma(A(1)),...,sigma(A(k))). whenever A(i) are nonnegative H-matrices, i.e. sigma(A(i)) greater than or equal to 0. The last inequality implies that the above function f` maps the set of nonnegative I-I-matrices into itself. In this note it is proven that these are the only continuous functions with this property. (C) 1998 Elsevier Science Inc. AU rights reserved.