The extremal case of some matrix inequalities (Q799756)

Let A and B be rectangular complex matrices and \(| A|\) and \(| B|\) the matrices of the absolut values of the entries of A and B, resp. Then, in general, the inequality \(| AB|\leq | A|| B|\) holds where the inequality sign is understood entrywise. Now, necessary conditions for the validity of the pure equal sign are derived. Further, let A denote a complex square matrix the spectral radius of which is less than 1 and \(<A>\) denote Ostrowski's comparison matrix. It is shown that \(| (I-A)^{-1}|\leq (I-| A|)^{-1}\) and conditions for the validity of the equal sign are given. Finally, let A be an H-matrix. It is wellknown that \(| A^{- 1}|\leq <A>^{-1}\). Conditions for this inequality to be an equality are given again.