A generalization of Bergstrom's inequality and some applications
The inequalities the authors considered are with respect to positive definite and semidefinite Hermitian matrices. They first derive a generalized Bergstrom's inequality and then use it to derive some other interesting results. The following notation is being used: \((\cdot)*=\) Hermitian transpose; \({\mathbf A}\odot{\mathbf B}=\) Schur-Hadamard (element by element) matrix product; \(A_{ij},({\mathbf A})_{ij}=i,j\) element of \({\mathbf A}\). Let \({\mathbf P},{\mathbf Q}\in{\mathcal C}^{N\times N}\) such that \({\mathbf P}>0\) and \({\mathbf Q}\geq 0\), \(Q_{ii}>0\). Then the authors prove the following tight inequalities: a) \(Q_{ii}[({\mathbf P}\odot{\mathbf Q})^{-1}]_{ii}\leq({\mathbf P}^{- 1})_{ii}\) ; b) \(Q_{ii}\{[\text{Re}({\mathbf P}\odot{\mathbf Q})]^{- 1}\}_{ii}\leq\max\{[\text{Re}({\Phi P\Phi}^*)]^{-1}\}_{ii}\) ; c) \(\text{det}({\mathbf P}\odot{\mathbf Q})\geq(\text{det}{\mathbf P})\prod^ N_{i=1}Q_{ii}\) ; d) \(\text{det}\text{Re}({\mathbf P}\odot{\mathbf Q})\geq\min\text{det}[\text{Re}({\Phi P\Phi}^*)]\prod^ N_{i=1}Q_{ii}\) , where in (b) and (d) maximization and minimization are over all unitary diagonal matrices \(\Phi\in{\mathcal C}^{N\times N}\). The paper contains the sections of generalization of Bergstrom's inequality, two Hadamard-product matrix inequalities, and generalization of Oppenheim's inequality.
scientific article; zbMATH DE number 32817