On relations among solutions of the Hermitian matrix equation $AXA^{*} = B$ and its three small equations

Abstract

Abstract. Assume that the linear matrix equation AXA * = B = B * has a Hermitian solution and is partitioned as. We study in this paper relations among the Hermitian solutions of the equation and the three small-size matrix equations A 1 X 1 A * 1 = B 11 , A 1 X 2 A * 2 = B 12 and A 2 X 3 A * 2 = B 22 . In particular, we establish closed-form formulas for calculating the maximal and minimal ranks and inertias of X − X 1 − X 2 − X * 2 − X 3 , and use the formulas to derive necessary and sufficient conditions for the Hermitian matrix equality X = X 1 + X 2 + X * 2 + X 3 to hold and Hermitian matrix inequalities X > ( , <, ) X 1 + X 2 + X * 2 + X 3 to hold in the Löwner partial ordering.