Optimization problems on the rank and inertia of the Hermitian matrix expression A−BX − (BX)∗ with applications
نویسنده
چکیده
We give in this paper some closed-form formulas for the maximal and minimal values of the rank and inertia of the Hermitian matrix expression A − BX ± (BX)∗ with respect to a variable matrix X. As applications, we derive the extremal values of the ranks/inertias of the matrices X and X ± X∗, where X is a (Hermitian) solution to the matrix equation AXB = C, respectively, and give necessary and sufficient conditions for the matrix equation AXB = C to have Hermitian, definite and Re-definite solutions. In particular, we derive the extremal ranks/inertias of Hermitian solutions X of the matrix equation AXA∗ = C, as well as the extremal ranks/inertias of Hermitian solution X of a pair of matrix equations A1XA ∗ 1 = C1 and A2XA ∗ 2 = C2. AMS Classifications: 15A09; 15A24; 15B57
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