On global optimizations of the rank and inertia of the matrix function
نویسنده
چکیده
For a given linear matrix function A1−B1XB 1 , where X is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations B2XB ∗ 2 = A2 and B3XB ∗ 3 = A3. As applications, we give necessary and sufficient conditions for the triple matrix equations B1XB ∗ 1 = A1, B2XB ∗ 2 = A2 and B3XB ∗ 3 = A3 to have a common Hermitian solution. In addition, we discuss the global optimizations on the rank and inertia of the common Hermitian solution of the pair of matrix equations B2XB ∗ 2 = A2 and B3XB ∗ 3 = A3. AMS subject classifications: 15A03, 15A09, 15A24; 65K10; 65K15
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Max-min optimizations on the rank and inertia of a linear Hermitian matrix expression subject to range, rank and definiteness restrictions
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