Index of Hadamard multiplication by positive matrices

Given a positive matrix A is an element of M-n(C), we introduce two notions of index of A, in terms of its action on the positive matrices by Hadamard multiplication: I-A = max{lambda greater than or equal to 0:\\A circle B\\ greater than or equal to lambda\\B\\, 0 less than or equal to B is an element of M-n}, IIA = max{lambda greater than or equal to 0:A circle B greater than or equa l to lambda B, 0 less than or equal to B is an element of M-n}. We also give formulae to compute these indices. A complete description is g iven in the 2 x 2 case. (C) 1999 Elsevier Science Inc. All rights reserved.