Rank and inertia optimizations of two Hermitian quadratic matrix functions subject to restrictions with applications
نویسندگان
چکیده
In this paper, we first give the maximal and minimal values of the ranks and inertias of the quadratic matrix functions q1(X) = Q1 − XP1X and q2(X) = Q2 − XP2X subject to a consistent matrix equation AX = B, where Q1, Q2, P1 and P2 are Hermitian matrices. As applications, we derive necessary and sufficient conditions for the solution of AX = B to satisfy the quadratic equality XP1X ∗ = Q1 and X P2X = Q2, as well as the quadratic inequalities XP1X ∗ > Q1 and XP2X > Q2 in the Löwner partial ordering. In particular, we give the minimal matrices of q1(X) and q2(X) subject to AX = B in the Löwner partial ordering. Mathematics Subject Classifications: 15A09; 15A24; 15A63; 15B10; 15B57; 65K10; 65K15
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تاریخ انتشار 2010