Index of Hadamard multiplication by positive matrices Original Research Article

Given a positive matrix Aset membership, variantMn(the field of complex numbers), we introduce two notions of index of A, in terms of its action on the positive matrices by Hadamard multiplication: IA=max{λgreater-or-equal, slanted0A○B;;greater-or-equal, slantedλ;B;, 0less-than-or-equals, slantBset membership, variantMn}, IIA=max{λgreater-or-equal, slanted0A○B;greater-or-equal, slantedλB, 0less-than-or-equals, slantBset membership, variantMn} We also give formulae to compute these indices. A complete description is given in the 2 × 2 case.