Analytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions and their applications
نویسنده
چکیده
A group of analytical formulas formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function φ(X) = (AXB + C )M(AXB + C) +D are established and their consequences are presented, where A, B, C and D are given complex matrices with A and C Hermitian. As applications, necessary and sufficient conditions for the two general quadratic matrix-valued functions ( k ∑ i=1 AiXiBi + C ) M ( k ∑ i=1 AiXiBi + C )∗ +D, k ∑ i=1 (AiXiBi + Ci )Mi(AiXiBi + Ci ) ∗ +D to be semi-definite are derived, respectively, where Ai, Bi, Ci, C, D, Mi and M are given matrices with Mi, M and D Hermitian, i = 1, . . . , k. Löwner partial ordering optimizations of the two matrix-valued functions are studied and their solutions are characterized. Mathematics Subject Classifications: 15A24; 15A63; 15B57; 65K10; 90C20; 90C22
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