The Hermitian solution to a matrix inequality under linear constraint

Abstract

In this paper, the necessary and sufficient conditions under which the matrix inequality $ C^*XC\geq D\ (>D) $ subject to the linear constraint $ A^*XA = B $ is solvable are deduced by means of the spectral decompositions of some matrices and the generalized singular value decomposition of a matrix pair. An explicit expression of the general Hermitian solution is also provided. One numerical example demonstrates the effectiveness of the proposed method.

Keywords

• matrix inequality
• matrix equation
• hermitian solution
• spectral decomposition
• generalized singular value decomposition